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```Projective and Polar Spaces
Peter J. Cameron
Queen Mary and Westfield College
2000
Preface
It is common now in academic circles to lament the decline in the teaching of
geometry in our schools and universities, and the resulting loss of “geometric intuition” among our students. On the other hand, recent decades have seen renewed
links between geometry and physics, to the benefit of both disciplines. One of the
world’s leading mathematicians has argued that the insights of “pre-calculus” geometry have a rôle to play at all levels of mathematical activity (Arnol’d [A]).
There is no doubt that a combination of the axiomatic and the descriptive approaches associated with algebra and geometry respectively can help avoid the
worst excesses of either approach alone.
These notes are about geometry, but by no means all or even most of geometry. I am concerned with the geometry of incidence of points and lines, over an
arbitrary field, and unencumbered by metrics or continuity (or even betweenness).
The major themes are the projective and affine spaces, and the polar spaces associated with sesquilinear or quadratic forms on projective spaces. The treatment
of these themes blends the descriptive (What do these spaces look like?) with the
axiomatic (How do I recognize them?) My intention is to explain and describe,
rather than to give detailed argument for every claim. Some of the theorems (especially the characterisation theorems) are long and intricate. In such cases, I give
a proof in a special case (often over the field with two elements), and an outline
of the general argument.
The classical works on the subject are the books of Dieudonné [L] and Artin [B].
I do not intend to compete with these books. But much has happened since
they were written (the axiomatisation of polar spaces by Veldkamp and Tits (see
Tits [S]), the classification of the finite simple groups with its many geometric
spin-offs, Buekenhout’s geometries associated with diagrams, etc.), and I have
included some material not found in the classical books.
Roughly speaking, the first five chapters are on projective spaces, the last
five on polar spaces. In more detail: Chapter 1 introduces projective and affine
i
ii
spaces synthetically, and derives some of their properties. Chapter 2, on projective
planes, discusses the rôle of Desargues’ and Pappus’ theorems in the coordinatisation of planes, and gives examples of non-Desarguesian planes. In Chapter 3,
we turn to the coordinatisation of higher-dimensional projective spaces, following Veblen and Young. Chapter 4 contains miscellaneous topics: recognition of
some subsets of projective spaces, including conics over finite fields of odd characteristic (Segre’s theorem); the structure of projective lines; and generation and
simplicity of the projective special linear groups. Chapter 5 outlines Buekenhout’s
approach to geometry via diagrams, and illustrates by interpreting the earlier characterisation theorems in terms of diagrams.
Chapter 6 relates polarities of projective spaces to reflexive sesquilinear forms,
and gives the classification of these forms. Chapter 7 defines polar spaces, the
geometries associated with such forms, and gives a number of these properties;
the Veldkamp–Tits axiomatisation and the variant due to Buekenhout and Shult
are also discussed, and proved for hyperbolic quadrics and for quadrics over the
2-element field. Chapter 8 discusses two important low-dimensional phenomena,
the Klein quadric and triality, proceeding as far as to define the polarity defining
the Suzuki–Tits ovoids and the generalised hexagon of type G2 . In Chapter 9, we
take a detour to look at the geometry of the Mathieu groups. This illustrates that
there are geometric objects satisfying axioms very similar to those for projective
and affine spaces, and also having a high degree of symmetry. In the final chapter,
we define spinors and use them to investigate the geometry of dual polar spaces,
especially those of hyperbolic quadrics.
The notes are based on postgraduate lectures given at Queen Mary and Westfield College in 1988 and 1991. I am grateful to members of the audience on these
occasions for their comments and especially for their questions, which forced me
to think things through more carefully than I might have done. Among many
pleasures of preparing these notes, I count two lectures by Jonathan Hall on his
beautiful proof of the characterisation of quadrics over the 2-element field, and
the challenge of producing the diagrams given the constraints of the typesetting
system!
In the introductory chapters to both types of spaces (Chapters 1 and 6), as well
as elsewhere in the text (especially Chapter 10), some linear algebra is assumed.
Often, it is necessary to do linear algebra over a non-commutative field; but the
differences from the commutative case are discussed. A good algebra textbook
(for example, Cohn (1974)) will contain what is necessary.
Peter J. Cameron, London, 1991
iii
Preface to the second edition
Materially, this edition is not very different from the first edition which was
published in the QMW Maths Notes series in 1991. I have converted the files
into LATEX, corrected some errors, and added some new material and a few more
references; this version does not represent a complete bringing up-to-date of the
original. I intend to publish these notes on the Web.
In the meantime, one important relevant reference has appeared: Don Taylor’s
book The Geometry of the Classical Groups [R]. (Unfortunately, it has already
gone out of print!) You can also look at my own lecture notes on Classical Groups
(which can be read in conjunction with these notes, and which might be integrated
with them one day). Other sources of information include the Handbook of Incidence Geometry [E] and (on the Web) two series of SOCRATES lecture notes at
http://dwispc8.vub.ac.be/Potenza/lectnotes.html
and
http://cage.rug.ac.be/˜fdc/intensivecourse2/final.html
Please note that, in Figure 2.3, there are a few lines missing: dotted lines utq
and urv and a solid line ub1 c2 . (The reason for this is hinted at in Exercise 3 in
Section 1.2.)
Peter J. Cameron, London, 2000
iv
Contents
1
Projective spaces
1.1 Fields and vector spaces . . . . . . . . . . . . . . .
1.2 Projective spaces . . . . . . . . . . . . . . . . . . .
1.3 The “Fundamental Theorem of Projective Geometry”
1.4 Finite projective spaces . . . . . . . . . . . . . . . .
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1
1
3
8
14
2
Projective planes
2.1 Projective planes . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Desarguesian and Pappian planes . . . . . . . . . . . . . . . . . .
2.3 Projectivities . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
19
22
27
3
Coordinatisation of projective spaces
3.1 The GF 2 case . . . . . . . . . .
3.2 An application . . . . . . . . . . .
3.3 The general case . . . . . . . . .
3.4 Lattices . . . . . . . . . . . . . .
3.5 Affine spaces . . . . . . . . . . .
3.6 Transitivity of parallelism . . . . .
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31
31
34
36
39
40
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45
45
48
51
57
59
62
4
Various topics
4.1 Spreads and translation planes . .
4.2 Some subsets of projective spaces
4.3 Segre’s Theorem . . . . . . . . .
4.4 Ovoids and inversive planes . . . .
4.5 Projective lines . . . . . . . . . .
4.6 Generation and simplicity . . . . .
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CONTENTS
vi
5 Buekenhout geometries
5.1 Buekenhout geometries . . . . . . . . . . . . . . . . . . . . . . .
5.2 Some special diagrams . . . . . . . . . . . . . . . . . . . . . . .
65
65
70
6 Polar spaces
6.1 Dualities and polarities . . . .
6.2 Hermitian and quadratic forms
6.3 Classification of forms . . . .
6.4 Classical polar spaces . . . . .
6.5 Finite polar spaces . . . . . .
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75
75
81
84
88
92
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97
97
101
105
107
109
113
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115
115
117
120
123
127
128
129
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133
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137
137
139
142
143
144
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7 Axioms for polar spaces
7.1 Generalised quadrangles . . . .
7.2 Diagrams for polar spaces . . .
7.3 Tits and Buekenhout–Shult . . .
7.4 Recognising hyperbolic quadrics
7.5 Recognising quadrics over GF 2
7.6 The general case . . . . . . . .
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8 The Klein quadric and triality
8.1 The Pfaffian . . . . . . . . . . . . .
8.2 The Klein correspondence . . . . .
8.3 Some dualities . . . . . . . . . . . .
8.4 Dualities of symplectic quadrangles
8.5 Reguli and spreads . . . . . . . . .
8.6 Triality . . . . . . . . . . . . . . .
8.7 An example . . . . . . . . . . . . .
8.8 Generalised polygons . . . . . . . .
8.9 Some generalised hexagons . . . . .
9 The geometry of the Mathieu groups
9.1 The Golay code . . . . . . . . .
9.2 The Witt system . . . . . . . . .
9.3 Sextets . . . . . . . . . . . . . .
9.4 The large Mathieu groups . . . .
9.5 The small Mathieu groups . . .
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CONTENTS
10 Exterior powers and Clifford algebras
10.1 Tensor and exterior products . . .
10.2 The geometry of exterior powers .
10.3 Near polygons . . . . . . . . . . .
10.3.1 Exercises . . . . . . . . .
10.4 Dual polar spaces . . . . . . . . .
10.5 Clifford algebras and spinors . . .
10.6 The geometry of spinors . . . . .
Index
vii
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147
147
150
152
155
155
156
159
168
1
Projective spaces
In this chapter, we describe projective and affine spaces synthetically, in terms of
vector spaces, and derive some of their geometric properties.
1.1
Fields and vector spaces
Fields will not necessarily be commutative; in other words, the term “field”
will mean “division ring” or “skew field”, while the word “commutative” will be
used where necessary. Often, though, I will say “skew field”, as a reminder. (Of
course, this refers to the multiplication only; addition will always be commutative.)
Given a field F, let
n : α F n α 0 n : n 1 0 F
Then I is an ideal in , hence I c for some non-negative integer c called the
I
characteristic of F. The characteristic is either 0 or a prime number. For each
value of the characteristic, there is a unique prime field which is a subfield of any
field of that characteristic: the rational numbers in characteristic zero, and the
integers modulo p in prime characteristic p.
Occasionally I will assume rudimentary results about field extensions, degree,
and so on.
Much of the time, we will be concerned with finite fields. The main results
about these are as follows.
Theorem 1.1 (Wedderburn’s Theorem) A finite field is commutative.
1
2
1. Projective spaces
Theorem 1.2 (Galois’ Theorem) A finite field has prime power order. For any
prime power q, there is a unique finite field of order q.
The unique field of order q is denoted by GF q . If q pd with p prime,
its additive structure is that of a d-dimensional vector space over its prime field
GF p (the integers modulo p). Its multiplicative group is cyclic (of order q 1),
and its automorphism group is cyclic (of order d). If d 1 (that is, if q is prime),
then GF q is the ring of integers mod q.
An anti-automorphism of a field is a bijection σ with the properties
c1 c2 σ
c1 c2 σ
cσ1 cσ2
cσ2 cσ1
The identity (or, indeed, any automorphism) is an anti-automorphism of a commutative field. Some non-commutative fields have anti-automorphisms. A wellknown example is the field of quaternions, with a basis over consisting of
elements 1 i j k satisfying
i2
j2
1
k2
ij
k
jk
i
ki
j;
the anti-automorphism is given by
a
bi
cj
Others, however, do not.
The opposite of the field F
ation is defined by the rule
dk
a
bi
cj
dk
, where the binary oper-
is the field F
c1 c2
c2 c1
Thus, an anti-automorphism of F is just an isomorphism between F and its opposite F .
For non-commutative fields, we have to distinguish between left and right
vector spaces. In a left vector space, if we write the product of the scalar c and the
vector v as cv, then c1 c2 v
c1 c2 v holds. In a right vector space, this condition
reads c1 c2 v
c2 c1 v. It is more natural to write the scalars on the right (thus:
vc), so that the condition is vc2 c1 v c2 c1 ). A right vector space over F is a
left vector space over F .
Our vector spaces will almost always be finite dimensional.
1.2. Projective spaces
3
For the most part, we will use left vector spaces. In this case, it is natural
to represent a vector by the row tuple of its coordinates with respect to some
basis; scalar multiplication is a special case of matrix multiplication. If the vector
space has dimension n, then vector space endomorphisms are represented by n n
matrices, acting on the right, in the usual way:
if v
vA ∑ viAi j
i
v1
vn .
The dual space V of a (left) vector space V is the set of linear maps from V
to F, with pointwise addition and with scalar multiplication defined by
fc v f cv
Note that this definition makes V a right vector space.
1.2
Projective spaces
A projective space of dimension n over a field F (not necessarily commutative!) can be constructed in either of two ways: by adding a hyperplane at infinity
to an affine space, or by “projection” of an n 1 -dimensional space. Both methods have their importance, but the second is the more natural.
Thus, let V be an n 1 -dimensional left vector space over F. The projective
space PG n F is the geometry whose points, lines, planes, . . . are the vector
subspaces of V of dimensions 1, 2, 3, . . . .
Note that the word “geometry” is not defined here; the properties which are
regarded as geometrical will emerge during the discussion.
Note also the dimension shift: a d-dimensional projective subspace (or flat)
is a d 1 -dimensional vector subspace. This is done in order to ensure that
familiar geometrical properties hold. For example, two points lie on a unique
line; two intersecting lines lie in a unique plane; and so on. Moreover, any ddimensional projective subspace is a d-dimensional projective space in its own
right (when equipped with the subspaces it contains).
To avoid confusion (if possible), I will from now on reserve the term rank (in
symbols, rk) for vector space dimension, so that unqualified “dimension” will be
geometric dimension.
A hyperplane is a subspace of codimension 1 (that is, of dimension one less
than the whole space). If H is a hyperplane and L a line not contained in H, then
H L is a point.
4
1. Projective spaces
A projective plane (that is, PG 2 F ) has the property that any two lines meet
3 and U W V with rk U
rk W
2,
in a (unique) point. For, if rk V
then U W V , and so rk U W
1; that is, U W is a point. From this, we
deduce:
Proposition 1.3 (Veblen’s Axiom) If a line intersects two sides of a triangle but
doesn’t contain their intersection, then it intersects the third side also.
!
! !
!!
!!
!!
!
Figure 1.1: Veblen’s Axiom
For the triangle is contained in a plane, and the hypotheses guarantee that the
line in question is spanned by points in the plane, and hence also lies in the plane.
Veblen’s axiom is sometimes called the Veblen-Young Axiom or Pasch’s Axiom. The latter name is not strictly accurate: Pasch was concerned with real projective space, and the fact that if two intersections are inside the triangle, the third
is outside; this is a property involving order, going beyond the incidence geometry
which is our concern here. In Section 3.1 we will see why 1.3 is referred to as an
“axiom”.
Another general geometric property of projective spaces is the following.
Proposition 1.4 (Desargues’ Theorem) In Figure 1.2, the three points p q r are
collinear.
In the case where the figure is not contained in a plane, the result is obvious
geometrically. For each of the three points p q r lies in both the planes a1 b1 c1 and
a2 b2 c2 ; these planes are distinct, and both lie in the 3-dimensional space spanned
by the three lines through o, and so their intersection is a line.
1.2. Projective spaces
5
o
& & ' ' '
" c& ( & ' '
" b ) " ) )& & ( '
"
"
"
"
& ) () (' '
"
"
"
&
" "
) ' a( )
"
"
&
"
'' ( ) )
"
"
&
"
"
( )\$
&&
"p# "# #
'
%
%
q\$ \$
\$
\$
# # # # #
%
\$
\$
a
\$
&
%
# # # # # \$ \$ \$ \$ \$ \$ & \$ \$ % %
b# # # #
# c&% %
1
1
1
r
2
2
2
Figure 1.2: Desargues’ Theorem
The case where the figure is contained in a plane can be deduced from the
“general” case as follows. Given a point o and a hyperplane H,write aa bb if
oaa obb are collinear triples and the lines ab and a b intersect in H (but none of
the points a a b b lies in H). Now Desargues’ Theorem is the assertion that the
relation is transitive. (For p q r are collinear if and only if every hyperplane
containing p and q also contains r; it is enough to assume this for the hyperplanes
not containing the points a a , etc.) So suppose that aa bb cc . The geometric argument of the preceding paragraph shows that aa cc if the configuration
is not coplanar; so suppose it is. Let od be a line not in this plane, with d H, and
choose d such that aa dd . Then bb dd , cc dd , and aa cc follow in
turn from the non-planar Desargues’ Theorem.
(If we are only given a plane initially, the crucial fact is that the plane can be
embedded in a 3-dimensional space.)
*
*
+
* *
*
*
*+
* *
*-+
*
/ /
*+
*+
*.+
* *+
*+
*
*
*
*
*.+
*
,
2 is not covered by this argument — can you see
Remark The case where F
why? — and, indeed, the projective plane over GF 2 contains no non-degenerate
Desargues configuration: it only contains seven points! Nevertheless, Desargues’
Theorem holds, in the sense that any meaningful degeneration of it is true in the
projective plane over GF 2 . We will not make an exception of this case.
It is also possible to prove Desargues’ Theorem algebraically, by choosing
6
1. Projective spaces
coordinates (see Exercise 1). However, it is important for later developments to
know that a purely geometric proof is possible.
Let V be a vector space of rank n 1 over F, and V its dual space. As we
saw, V is a right vector space over F, and so can be regarded as a left vector
space over the opposite field F . It has the same rank as V if this is finite. Thus
we have projective spaces PG n F and PG n F , standing in a dual relation to
one another. More precisely, we have a bijection between the flats of PG n F
and those of PG n F , given by
U 1
0
f V : u U 2 fu
Ann U
0
This correspondence preserves incidence and reverses inclusion:
4 Ann U1 5 Ann U2 Ann U2 0
Moreover, the (geometric) dimension of Ann U is n 1 dim U .
U1 U2
Ann U1 U2
Ann U1 U2
3
Ann U2
Ann U1
Ann U1
This gives rise to a duality principle, where any configuration theorem in projective space translates into another (over the opposite field) in which inclusions
are reversed and dimensions suitably modified. For example, in the plane, the dual
of the statement that two points lie on a unique line is the statement that two lines
meet in a unique point.
We turn briefly to affine spaces. The description closest to that of projective
spaces runs as follows. Let V be a vector space of rank n over F. The points, lines,
planes, . . . of the affine space AG n F are the cosets of the vector subspaces of
rank 0, 1, 2, . . . . (No dimension shift this time!) In particular, points are cosets of
the zero subspace, in other words, singletons, and we can identify them with the
vectors of V . So the affine space is “a vector space with no distinguished origin”.
The other description is: AG n F is obtained from PG n F by deleting a
hyperplane together with all the subspaces it contains.
The two descriptions are matched up as follows. Take the vector space
V
Fn
6
1
x x x 0 1
n
: x0
xn F Let W be the hyperplane defined by the equation x0 0. The points remaining
are rank 1 subspaces spanned by vectors with x0 0; each point has a unique
spanning vector with x0 1. Then the correspondence between points in the two
descriptions is given by
7
,
1 x1 xn 81 x1 xn 0
1.2. Projective spaces
7
(See Exercise 2.)
In AG n F , we say that two subspaces are parallel if (in the first description)
they are cosets of the same vector subspace, or (in the second description) they
have the same intersection with the deleted hyperplane. Parallelism is an equivalence relation. Now the projective space can be recovered from the affine space
as follows. To each parallel class of d-dimensional subspaces of AG n F corresponds a unique d 1 -dimensional subspace of PG n 1 F . Adjoin to the
affine space the points (and subspaces) of PG n 1 F , and adjoin to all members of a parallel class all the points in the corresponding subspace. The result is
PG n F .
The distinguished hyperplane is called the hyperplane at infinity or ideal hyperplane. Thus, an affine space can also be regarded as “a projective space with a
distinguished hyperplane”.
The study of projective geometry is in a sense the outgrowth of the Renaissance theory of perspective. If a painter, with his eye at the origin of Euclidean 3space, wishes to represent what he sees on a picture plane, then each line through
the origin (i.e., each rank 1 subspace) should be represented by a point of the picture plane, viz., the point at which it intersects the picture plane. Of course, lines
parallel to the picture plane do not intersect it, and must be regarded as meeting it
in ideal “points at infinity”. Thus, the physical picture plane is an affine plane, and
is extended to a projective plane; and the points of the projective plane are in oneto-one correspondence with the rank 1 subspaces of Euclidean 3-space. It is easily
checked that lines of the picture plane correspond to rank 2 subspaces, provided
we make the convention that the points at infinity comprise a single line. Not that
the picture plane really is affine rather than Euclidean; the ordinary distances in it
do not correspond to distances in the real world.
Exercises
1. Prove Desargues’ Theorem in coordinates.
2. Show that the correspondence defined in the text between the two descriptions of affine space is a bijection which preserves incidence, dimension, and parallelism.
3. The LATEX typesetting system provides facilities for drawing diagrams. In
a diagram, the slope of a line is restricted to being infinity or a rational number
whose numerator and denominator are each at most 6 in absolute value.
(a) What is the relation between the slopes of the six lines of a complete
quadrangle (all lines joining four points)? Investigate how such a figure can be
8
1. Projective spaces
drawn with the above restriction on the slopes.
(b) Investigate similarly how to draw a Desargues configuration.
1.3
The “Fundamental Theorem of Projective Geometry”
An isomorphism between two projective spaces is a bijection between the
point sets of the spaces which maps any subspace into a subspace (when applied in
either direction). A collineation of PG n F is an isomorphism from PG n F to
itself. The theorem of the title of this section has two consequences: first, that isomorphic projective spaces have the same dimension and the same coordinatising
field; second, a determination of the group of all collineations.
We must assume that n 1; for the only proper subspaces of a projective line
are its points, and so any bijection is an isomorphism, and the collineation group
is the full symmetric group. (There are methods for assigning additional structure
to a projective line, for example, using cross-ratio; these will be discussed later
on, in Section 4.5.)
The general linear group GL n 1 F is the group of all non-singular linear
transformations of V F n 1 ; it is isomorphic to the group of invertible n 1
n 1 matrices over F. (In general, the determinant is not well-defined, so we
cannot identify the invertible matrices with those having non-zero determinant.)
Any element of GL n 1 F maps subspaces of V into subspaces of the same
rank, and preserves inclusion; so it induces a collineation of PG n F . The group
Aut F of automorphisms of F has a coordinate-wise action on V n 1 ; these transformations also induce collineations. The group generated by GL n 1 F and
Aut F (which is actually their semi-direct product) is denoted by ΓL n 1 F ;
its elements are called semilinear transformations. The groups of collineations of
PG n F induced by GL n 1 F and ΓL n 1 F are denoted by PGL n 1 F
and PΓL n 1 F , respectively.
More generally, a semi-linear transformation from one vector space to another
is the composition of a linear transformation and a coordinate-wise field automorphism of the target space.
9
6
:
6
Theorem 1.5 (Fundamental Theorem of Projective Geometry) Any isomorphism
between projective spaces of dimension at least 2 is induced by a semilinear transformation between the underlying vector spaces, unique up to scalar multiplication.
1.3. The “Fundamental Theorem of Projective Geometry”
9
Before outlining the proof, we will see the two important corollaries of this
result. Both follow immediately from the theorem (in the second case, by taking
the two projective spaces to be the same).
Corollary 1.6 Isomorphic projective spaces of dimension at least 2 have the same
dimension and are coordinatised by isomorphic fields.
Corollary 1.7 (a) For n
PΓL n 1 F .
9
1, the collineation group of PG n F is the group
(b) The kernel of the action of ΓL n 1 F on PG n F is the group of non-zero
scalars (acting by left multiplication).
Remark The point of the theorem, and the reason for its name, is that the algebraic structure of the underlying vector space can be recovered from the incidence
geometry of the projective space. The proof is a good warm-up for the coordinatisation theorems I will be discussing soon. In fact, the proof concentrates
on Corollary 1.7, for ease of exposition. The dimension of a projective space is
two less than the number of subspaces in a maximal chain (under inclusion); and
our argument shows that the geometry determines the coordinatising field up to
isomorphism.
Proof We show first that two semi-linear transformations which induce the same
collineation differ only by a scalar factor. By following one by the inverse of
the other, we see that it suffices to show that a semi-linear transformation which
fixes every point of PG n F is a scalar multiplication. So let v vσ A fix every
point of PG n F , where σ Aut F and A GL n 1 F . Then every vector
is mapped to a scalar multiple of itself. Let e0
en be the standard basis for V .
Then (since σ fixes the standard basis vectors) we have ei A λi ei for i 0
n.
Also,
e0 ;
en A
; λ0 e0
λ e0
so λ0
λn λ.
Now, for any µ F, the vector 1 µ 0
so we have λµ µσ λ. Thus
; 0
vσ A
vσ λ
λn en
en
say,
; 0 ;
is mapped to the vector λ µσλ 0
λv
10
1. Projective spaces
for any vector v, as required.
Note that the field automorphism σ is conjugation by the element λ (that is,
µσ λµλ 1 ); in other words, an inner automorphism.
Now we prove that any isomorphism is semilinear. The strategy is similar.
Call an n 2 tuple of points special if no n 1 of them are linearly dependent.
We have:
<
There is a linear map carrying any special tuple to any other (in the
same space, or another space of the same dimension over the same
field).
(For, given a special tuple in the first space, spanning vectors for the first n 1
points form a basis e0
en , and the last point is spanned by a vector with all
coordinates non-zero relative to this basis. Adjusting the basis vectors by scalar
factors, we may assume that the last point is spanned by e0
en . Similarly,
the points of a special tuple in the second space are spanned by the vectors of a
basis f0
fn , and f0
fn . The unique linear transformation carrying the
first basis to the second also carries the first special tuple to the second.)
Let θ be any isomorphism. Then there is a linear map φ which mimics the
effect of θ on a special n 2 -tuple. Composing θ with the inverse of φ, we
obtain an automorphism of PG n F which fixes the n 2 -tuple pointwise. We
have to show that such an automorphism is the product of a scalar and a field
automorphism. (Note that, as we saw above, left and right multiplications by λ
differ by an inner automorphism.)
We assume that n 2; this simplifies the argument, while retaining its essential
features. So let g be a collineation fixing the spans of e0 e1 e2 and e0 e1 e2 . We
use homogeneous coordinates, writing these vectors as 1 0 0 , 0 1 0 , 0 0 1 ,
and 1 1 1 , and denote the general point by x y z .
The points on the line x0 0 x2 , apart from 1 0 0 , have the form x 0 1
for x F, and so can be identified with elements of F. Now the bijection between this set and the set of points 0 y 1 on the line 0 x1 x2 , given by
x01
0 x 1 , can be geometrically defined in a way which is invariant under collineations fixing the four reference points (see Fig. 1.3). The figure also
shows that the coordinates of all points in the plane are determined.
Furthermore, the operations of addition and multiplication in F can be defined
geometrically in the same sense (see Figures 1.4 and 1.6). (The definitions look
more familiar if we take the line x1 x2 0 to be at infinity, and draw the figure
in the affine plane with lines through 1 0 0 and 0 1 0 horizontal and vertical
respectively. this has been done for addition in Figure 1.5; the reader should draw
> =
1.3. The “Fundamental Theorem of Projective Geometry”
C 0?D!' 1 D 0E
'' !? ! ?
'' ! ! ?
'' !
''
C 0 D x D 1B E B '
B'' B
''
'
C 0 D 1 D 1A E A A
@@ @
C 0 D 0 D 1E
? ?
!! ? ?
!! ? ?
? ? C
!!
B B ! @ @ ? 1 D 1 D 0E
B B ! @
?
B@ @ ! B C x D x D 1E ? ?
'' @ ! !B B
'' @ @
!! B B B B
A @ A' C 1A D 1AD 1E A A
A A A ! A! A A
''
!! A
'
C 1 D 0 D 1E
C x D 0 D 1E
11
? ?
B B ? ?
B B ? ?
A A A A BA BA ?B
A C 1 D 0 D 0E
Figure 1.3: Bijection between the axes
the corresponding diagram for multiplication.) It follows that any collineation
fixing our four basic points induces an automorphism of the field F, and its actions
on the coordinates agree. The theorem is proved.
A group G acting on a set Ω is said to be t-transitive if, given any two t-tuples
α1
αt and β1
βt of distinct elements of Ω, some element of G carries
the first tuple to the second. G is sharply t-transitive if there is a unique such
element. (If the action is not faithful, it is better to say: two elements of G which
agree on t distinct points of Ω agree everywhere.)
Since any two distinct points of PG n F are linearly independent, we see that
PΓL n 1 F (or even PGL n 1 F ) is 2-transitive on the points of PG n F . It
is never 3-transitive (for n 1); for some triples of points are collinear and others
are not, and no collineation can map one type to the other.
; ; 9
12
1. Projective spaces
C 0 D?GF 1 D 0E
F GF ? G ?
FF G G ? ?
FF G G ? ?
FF G G ? ?
? ?
FF G G
? C
FF G G
@@ @ H H ? 1 D 1? D 0E
FF G G @ H ?
F F @ @ G G HH
G G HH
# # # # # @ F F @C
# y#D yD 1# E # G H C
@
F
# H # G x# 6 y# D yD 1# E
@@
FF
HH G G #
@
F
C 0 D 0 D 1E
C yD 0 D 1E C x D 0 D 1C E x6 yD 0 D 1E
? ?
? ?
? ?
# # # # # ? ?
# # #C
1 D 0 D 0E
Figure 1.4: Addition
I will digress here to describe the analogous situation for PG 1 F , even
though the FTPG does not apply in this case.
Proposition 1.8 (a) The group PGL 2 F is 3-transitive on the points of PG 1 F ,
and is sharply 3-transitive if and only if F is commutative.
(b) There exist skew fields F for which the group PGL 2 F is 4-transitive on
PG 1 F .
Proof The first part follows just as in the proof of the FTPG, since any three
points of PG 1 F have the property that no two are linearly dependent. Again,
as in that theorem, the stabiliser of the three points with coordinates 1 0 , 0 1
and 1 1 is the group of inner automorphisms of F, and so is trivial if and only if
F is commutative.
1.3. The “Fundamental Theorem of Projective Geometry”
C
@@ @ yD yE @
@
@
C 0 D 0 E C yD 0 E C x D 0 E
13
C
@ @ x 6 yD y E
C x 6 yD 0 E
Figure 1.5: Affine addition
There exist skew fields F with the property that any two elements different
from 0 and 1 are conjugate in the multiplicative group of F. Clearly these have
the required property. (This fact is due to P. M. Cohn ; it is established by
a construction analogous to that of Higman, Neumann and Neumann  for
groups. Higman et al. used their construction to show that there exist groups
in which all non-identity elements are conjugate; Cohn’s work shows that there
are multiplicative groups of skew fields with this property. Note that such a field
has characteristic 2. For, if not, then 1 1 0, and any automorphism must fix
1 1.)
,
Finally, we consider collineations of affine spaces.
Parallelism in an affine space has an intrinsic, geometric definition. For two dflats are parallel if and only if they are disjoint and some d 1 -flat contains both.
It follows that any collineation of AG n F preserves parallelism. The hyperplane
at infinity can be constructed from the parallel classes (as we saw in Section 1.2);
so any collineation of AG n F induces a collineation of this hyperplane, and
hence of the embedding PG n F . Hence:
Theorem 1.9 The collineation group of AG n F is the stabiliser of a hyperplane
in the collineation group of PG n F .
Using this, it is possible to determine the structure of this group for n
Exercise 2).
9
9
1 (see
Proposition 1.10 For n 1, the collineation group of AG n F is the semi-direct
product of the additive group of F n and ΓL n F .
This group is denoted by AΓL n F . The additive group acts by translation,
and the semilinear group in the natural way.
14
1. Projective spaces
C 0 D?M(L' 1 D 0E
' L' (L ? M ( M ?
' ' L L ( M (? M ?
K K' L ( M ?
' ' K L L K ( (M M? ? C
' ' L L K K ( ( M I I M ? 1 D ? yD 0 E
' ' L L I K I ( K C x( DK xM yDM 1E ? ? C
' ' L L I I ( ( K @ M K @ @M C xyD ?x1yD D 1?1D E 0E
'' I I L L @ @ ( ( K M K M K ? ?
J J J JI' C L @
K ?
II ' J @1 D yJ @ D 1L JE C yJ D yD 1J E ( ( M M K K ? ?
II @ @ ' ' C 1 D 1 D 1L E L J J J J ( ( J J M M K K ?K ?
( ( J J M JM K
II@ @ ' ' ' L L L
( MJ J J
C 0 D 0 D 1 E C 1 D 0 D 1 E C yD 0 D 1 E
C x D 0 D 1E C xyD 0 D 1E
?K ?
J KJ C
1 D 0 D 0E
Figure 1.6: Multiplication
Exercises
9
1. Prove the FTPG for n 2.
2. Use the correspondence between the two definitions of AG n F given in
the last section to deduce Proposition 1.10 from Theorem 1.9.
1.4
Finite projective spaces
Over the finite field GF q , the n-dimensional projective and affine spaces
and their collineation groups are finite, and can be counted. In this section, we
display some of the relevant formulæ. We abbreviate PG n GF q to PG n q ,
and similarly for affine spaces, collineation groups, etc.
A vector space of rank n over GF q is isomorphic to GF q n , and so the
number of vectors is qn . In consequence, the number of vectors outside a subspace
1.4. Finite projective spaces
of rank k is qn
15
qk .
Proposition 1.11 The number of subspaces of rank k in a vector space of rank n
over GF q is
qn 1 qn q
qn qk 1
qk 1 qk q
qk qk 1
N 5;O N 5 O < < Remark This number is called a Gaussian coefficient, and is denoted by
P nkQ q.
Proof First we count the number of choices of k linearly independent vectors.
The ith vector may be chosen arbitrarily outside the subspace of rank i 1 spanned
by its predecessors, hence in qn qi 1 ways. Thus, the numerator is the required
number of choices.
Now any k linearly independent vectors span a unique subspace of rank k; so
the number of subspaces is found by dividing the number just calculated by the
number of choices of a basis for a space of rank k. But the latter is given by the
same formula, with k replacing n.
<
qn 1N qn q5 O qn Proposition 1.12 The order of GL n q is
qn
< 1 0 The order of ΓL n q is the above number multiplied by d, where q pd with p
prime; and the orders of PGL n q and PΓL n q are obtained by dividing these
numbers by q 1 .
Proof An element of GL n q is uniquely determined by the image of the standard basis, which is an arbitrary basis of GF q n; and the proof of Proposition 1.11
shows that the number of bases is the number quoted. The remainder of the proposition follows from the remarks in Section 1.3, since GF q has q 1 non-zero
scalars, and its automorphism group has order d.
The formula for the Gaussian coefficient makes sense, not just for prime power
values of q, but for any value of q different from 1. There is a combinatorial
interpretation for any integer q 1 (Exercise 3). Moreover, by l’Hôpital’s rule,
limq 1 qa 1 qb 1
a b; it follows that
R ST S
9
R
lim
q 1
Un
k
V
XW
q
n
k
Y 16
1. Projective spaces
This illustrates just one of the many ways in which subspaces of finite vector
spaces resemble subsets of sets.
It follows immediately from Propopsition 1.11 that the numbers of k-dimensional
flats in PG n q and AG n q are nk 11 q and qn k nk q respectively.
Projective and affine spaces provide important examples of designs, whose
parameters can be expressed in terms of the Gaussian coefficients.
A t-design with parameters v k λ , or t- v k λ design, consists of a set X
of v points, and a collection of k-element subsets of X called blocks, with the
property that any t distinct points of X are contained in exactly λ blocks. Designs
were first used by statisticians, such as R. A. Fisher, for experimental design (e.g.
to facilitate analysis of variance). The terms “design” and “block”, and the letter
v (the initial letter of “variety”), reflect this origin.
P 66 Q
< PQ
Z
Proposition 1.13 (a) The points and m-dimensional flats in PG n q form a 2design with parameters
[ U
n
Um
U n
1
1
1 V q m 1V q \
(b) The points and m-dimensional flats of AG n q form a 2-design with parameters
[
U n 1
n
m
q q m 1V q \
If q 2, then it is a 3-design, with λ P mn < 22Q 2 .
<
1
V q
1
Proof The values of v and k are clear in both cases.
(a) Let V be the underlying vector space of rank n 1. We want to count
the subspaces of rank m 1 containing two given rank 1 subspaces P1 and P2 . If
L P1 P2 , then L has rank 2, and a subspace contains P1 and P2 if and only if it
contains L. Now, by the Third Isomorphism Theorem, the rank m 1 subspaces
containing L are in 1-1 correspondence with the rank m 1 subspaces of the rank
n 1 space V L.
(b) In AG n q , to count subspaces containing two points, we may assume
(by translation) that one of the points is the origin. An affine flat containing the
origin is a vector subspace, and a subspace contains a non-zero vector if and only
if it contains the rank 1 subspace it spans. The result follows as before. In the
case when q 2, a rank 1 subspace contains only one non-zero vector, so any two
distinct non-zero vectors span a rank 2 subspace.
S
1.4. Finite projective spaces
17
Remark The essence of the proof is that the quotient of either PG n q or
AG n q by a flat F of dimension d is PG n d 1 q . (The flats of the quotient
space are precisely the flats of the original space containing F.) This assertion is
true over any field at all, and lies at the basis of an approach to geometry which
we will consider in Chapter 5.
An automorphism of a design is a permutation of the points which maps any
block to a block.
]
]
Proposition 1.14 For 0 m n, the design of points and m-dimensional flats in
PG n q or AG n q is PΓL n 1 q or AΓL n 1 q respectively, except in the
affine case with q 2 and m 1.
Proof By the results of Section 1.3, it suffices to show that the entire geometry
can be recovered from the points and m-dimensional flats. This follows immediately from two observations:
(a) the unique line containing two points is the intersection of all the m-dimensional
flats containing them;
(b) except for affine spaces over GF 2 , a set of points is a flat if and only if it
contains the line through any two of its points.
Affine spaces over GF 2 are exceptional: lines have just two points, and any two
points form a line. However, analogous statements hold for planes: three points
lie in a unique plane, and we have
(aa) the plane through three points is the intersection of all the flats of dimension m which contain them (for m 1);
9
(bb) a set of points is a flat if and only if it contains the plane through any three
of its points.
The proofs are left as exercises.
Exercises
1. Prove the assertions (a), (b), (aa), (bb) in Proposition 1.14.
2. Prove that the probability that a random n n matrix over a given finite field
1.
GF q is non-singular tends to a limit c q as n ∞, where 0 c q
] 4]
18
1. Projective spaces
3. Prove that the total number F n of subspaces of a vector space of rank n
over a given finite field GF q satisfies the recurrence
1
F n
qn 1 F n 10 2F n
4. Let S be an “alphabet” of size q, with two distinguished elements 0 and 1
(but not necessarily a finite field). A k n matrix with entries from S is (as usual)
in reduced echelon form if
^
^
it has no zero rows;
^
the first non-zero entry in any row is a 1;
^
the “leading 1s” in later rows occur further to the right;
the other entries in the column of a “leading 1” are all 0.
Prove that the number of k n matrices in reduced echelon form is
detail in the case n 4, k 2.
5. Use the result of Exercise 4 to prove the recurrence relation
Un
k
V
q
q
n
U n
1
k
U n 1
V q k 1V q P nkQ q. Verify in
2
Projective planes
Projective and affine planes are more than just spaces of smallest (non-trivial)
dimension: as we will see, they are truly exceptional, and also they play a crucial
rôle in the coordinatisation of arbitrary spaces.
2.1
Projective planes
We have seen in Sections 1.2 and 1.3 that, for any field F, the geometry
PG 2 F has the following properties:
(PP1) Any two points lie on exactly one line.
(PP2) Any two lines meet in exactly one point.
(PP3) There exist four points, no three of which are collinear.
I will now use the term projective plane in a more general sense, to refer to any
structure of points and lines which satisfies conditions (PP1)-(PP3) above.
In a projective plane, let p and L be a point and line which are not incident.
The incidence defines a bijection between the points on L and the lines through p.
By (PP3), given any two lines, there is a point incident with neither; so the two
lines contain equally many points. Similarly, each point lies on the same number
of lines; and these two constants are equal. The order of the plane is defined to
be one less than this number. The order of PG 2 F is equal to the cardinality of
F. (We saw in the last section that a projective line over GF q has 21 q q 1
points; so PG 2 q is a projective plane of order q. In the infinite case, the claim
follows by simple cardinal arithmetic.)
19
20
2. Projective planes
Given a finite projective plane of order n, each of the n 1 lines through a point
p contains n further points, with no duplications, and all points are accounted for
in this way. So there are n2 n 1 points, and the same number of lines. The
points and lines form a 2- n2 n 1 n 1 1 design. The converse is also true
(see Exercise 2).
Do there exist projective planes not of the form PG 2 F ? The easiest such
examples are infinite; I give two completely different ones below. Finite examples
will appear later.
Example 1: Free planes. Start with any configuration of points and lines having
the property that two points lie on at most one line (and dually), and satisfying
(PP3). Perform the following construction. At odd-numbered stages, introduce
a new line incident with each pair of points not already incident with a line. At
even-numbered stages, act dually: add a new point incident with each pair of
lines for which such a point doesn’t yet exist. After countably many stages, a
projective plane is obtained. For given any two points, there will be an earlier
stage at which both are introduced; by the next stage, a unique line is incident with
both; and no further line incident with both is added subsequently; so (PP1) holds.
Dually, (PP2) holds. Finally, (PP3) is true initially and remains so. If we start
with a configuration violating Desargues’ Theorem (for example, the Desargues
configuration with the line pqr “broken” into separate lines pq, qr, rp), then the
resulting plane doesn’t satisfy Desargues’ Theorem, and so is not a PG 2 F .
Example 2: Moulton planes. Take the ordinary real affine plane. Imagine that
the lower half-plane is a refracting medium which bends lines of positive slope
so that the part below the axis has twice the slope of the part above, while lines
with negative (or zero or infinite) slope are unaffected. This is an affine plane, and
has a unique completion to a projective plane (see later). The resulting projective
plane fails Desargues’ theorem. To see this, draw a Desargues configuration in the
ordinary plane in such a way that just one of its ten points lies below the axis, and
just one line through this point has positive slope.
The first examples of finite planes in which Desargues’ Theorem fails were
constructed by Veblen and Wedderburn . Many others have been found since,
but all known examples have prime power order. The Bruck–Ryser Theorem 
asserts that, if a projective plane of order n exists, where n 1 or 2 (mod 4), then
n must be the sum of two squares. Thus, for example, there is no projective plane
of order 6 or 14. This theorem gives no information about 10, 12, 15, 18, . . . .
2.1. Projective planes
21
Recently, Lam, Swiercz and Thiel  showed by an extensive computation that
there is no projective plane of order 10. The other values mentioned are undecided.
An affine plane is an incidence structure of points and lines satisfying the
following conditions (in which two lines are called parallel if they are equal or
disjoint):
(AP1) Two points lie on a unique line.
(AP2) Given a point p and line L, there is a unique line which contains p and is
parallel to L.
(AP3) There exist three non-collinear points.
Remark. Axiom (AP2) for the real plane is an equivalent form of Euclid’s “parallel postulate”. It is called “Playfair’s Axiom”, although it was stated explicitly
by Proclus.
Again it holds that AG 2 F is an affine plane. More generally, if a line and all
its points are removed from a projective plane, the result is an affine plane. (The
removed points and line are said to be “at infinity”. Two lines are parallel if and
only if they contain the same point at infinity.
Conversely, let an affine plane be given, with point set and line set . It
follows from (AP2) that parallelism is an equivalence relation on . Let be the
set of equivalence classes. For each line L
, let L
L Q , where Q is the
parallel class containing L. Then the structure with point set
, and line set
L :L
, is a projective plane. Choosing as the line at infinity, we
recover the original affine plane.
We will have more to say about affine planes in Section 3.5.
Exercises
1. Show that a structure which satisfies (PP1) and (PP2) but not (PP3) must be
of one of the following types:
(a) There is a line incident with all points. Any further line is a singleton,
repeated an arbitrary number of times.
(b) There is a line incident with all points except one. The remaining lines all
contain two points, the omitted point and one of the others.
2. Show that a 2- n2 n 1 n 1 1 design (with n 1) is a projective plane
of order n.
3. Show that, in a finite affine plane, there is an integer n 1 such that
22
2. Projective planes
every line has n points;
every point lies on n
1 lines;
there are n2 points;
there are n
1 parallel classes with n lines in each.
(The number n is the order of the affine plane.)
4. (The Friendship Theorem.) In a finite society, any two individuals have a
unique common friend. Prove that there exists someone who is everyone else’s
friend.
[Let X be the set of individuals,
F x : x X , where F x is the set
of friends of X . Prove that, in any counterexample to the theorem, X
is a
projective plane, of order n, say.
Now let A be the real matrix of order n2 n 1, with x y entry 1 if x and y
are friends, 0 otherwise. Prove that
A2
nI
J
where I is the identity matrix and J the all-1 matrix. Hence show that the real
symmetric matrix A has eigenvalues n 1 (with multiplicity 1) and
n. Using
the fact that A has trace 0, calculate the multiplicity of the eigenvalue n, and
hence show that n 1.]
5. Show that any Desargues configuration in a free projective plane must lie
within the starting configuration. [Hint: Suppose not, and consider the last point
or line to be added.]
2.2
Desarguesian and Pappian planes
It is no coincidence that we distinguished the free and Moulton planes from
PG 2 F s in the last section by the failure of Desargues’ Theorem.
Theorem 2.1 A projective plane is isomorphic to PG 2 F for some F if and only
if it satisfies Desargues’ Theorem.
I do not propose to give a detailed proof of this important result; but some
comments on the proof are in order.
We saw in Section 1.3 that, in PG 2 F , the field operations (addition and multiplication) can be defined geometrically, once a set of four points with no three
2.2. Desarguesian and Pappian planes
23
collinear has been chosen. By (PP3), such a set of points exists in any projective
plane. So it is possible to define two binary operations on a set consisting of a
line with a point removed, and to coordinatise the plane with this algebraic object. Now it is obvious that any field axiom translates into a certain “configuration
theorem”, so that the plane is a PG 2 F if and only if all these “configuration
theorems” hold. What is not obvious, and quite remarkable, is that all these “configuration theorems” follow from Desargues’ Theorem.
Another method, more difficult in principle but much easier in detail, exploits
the relation between Desargues’ Theorem and collineations.
Let p be a point and L a line. A central collineation with centre p and axis
L is a collineation fixing every point on L and every line through p. It is called
an elation if p is on L, a homology otherwise. The central collineations with
centre p and axis L form a group. The plane is said to be p L -transitive if this
group permutes transitively the set M p L M for any line M L on p (or,
equivalently, the set of lines on q different from L and pq, where q p is a point
of L).
o
# #& & '' '
# & '
! c& & ( ( ' '
&
& ) ) () ( ' '
#
#
!
! ! # #! ) )
!
!
! ! !
#
!
&
!
) ' a( )
!
#
!
&
!
'( ) )
!
#
!
&
!
' ( )
!
#
&
"p! !" " "
#
& \$ \$ \$ % \$ '% a%\$ q\$ \$ \$
" " " "
#
" " " " #\$ \$ \$ \$ \$ \$ & &\$ \$ % % %
" b" " "
" " c&% %
b1
1
1
r
2
2
2
Figure 2.1: The Desargues configuration
Theorem 2.2 A projective plane satisfies Desargues’ Theorem if and only if it is
p L -transitive for all points p and lines L.
24
2. Projective planes
Proof Let us take another look at the Desargues configuration (Fig. 2.1). It is
clear that any central configuration with centre o and axis L which carries a1 to a2
is completely determined at every point b1 not on M. (The line a1 a2 meets L at a
fixed point r and is mapped to b1 b2 ; so b2 is the intersection of ra2 and ob1 .) Now,
if we replace M with another line M through o, we get another determination of
the action of the collineation. It is easy to see that the condition that these two
specifications agree is precisely Desargues’ Theorem.
*
The proof shows a little more. Once the action of the central collineation on
one point of M o L M is known, the collineation is completely determined.
So, if Desargues’ Theorem holds, then these groups of central collineations act
sharply transitively on the relevant set.
Now the additive and multiplicative structures of the field turn up as groups
of elations and homologies respectively with fixed centre and axis. We see immediately that these structures are both groups. More of the axioms are easily
deduced too. For example, let L be a line, and consider all elations with axis L
(and arbitrary centre on L). This set is a group G. For each point p on L, the
elations with centre p form a normal subgroup. These normal subgroups partition
the non-identity elements of G, since a non-identity elation has at most one centre. But a group having such a partition is abelian (see Exercise 2). So addition is
commutative.
In view of this theorem, projective planes over skew fields are called Desarguesian planes.
There is much more to be said about the relationships among configuration
theorems, coordinatisation, and central collineations. I refer to Dembowski’s book
for some of these. One such relation is of particular importance.
Pappus’ Theorem is the assertion that, if alternate vertices of a hexagon are
collinear (that is, the first, third and fifth, and also the second, fourth and sixth),
then also the three points of intersection of opposite edges are collinear. See
Fig. 2.2.
+ Theorem 2.3 A projective plane satisfies Pappus’ Theorem if and only if it is
isomorphic to PG 2 F for some commutative field F.
Proof The proof involves two steps. First, a purely geometric argument shows
that Pappus’ Theorem implies Desargues’. This is shown in Fig. 2.3. This figure shows a potential Desargues configuration, in which the required collinearity
is shown by three applications of Pappus’ Theorem. The proof requires four new
2.2. Desarguesian and Pappian planes
2 2- 2- / /
2
2
2
- /
2 , , 2 1 12
2
2
2
20( 0
/
( 0 , , 1- 1 - /
( 0 ,0 - 1 /
(, 01//
(, - 0
1
, -( ( 0 0 0 / / 1 1
, - (
/ 0
,(
/
0 0 1
-,
(
/
0
. . . . . . (/
. . . . . .
.
25
1
0 101
. . . 0. 1
Figure 2.2: Pappus’ Theorem
points, s a1 b1 a2 c2 , t b1 c1 os, u b1 c2 oa1 , and v b2 c2 os. Now Pappus’ Theorem, applied to the hexagon osc2 b1 c1 a1 , shows that q u t are collinear;
applied to osb1 c2 b2 a2 , shows that r u v are collinear; and applied to b1tuvc2s (using the two collinearities just established), shows that p q r are collinear. The
derived collinearities are shown as dotted lines in the figure. (Note that the figure
shows only the generic case of Desargues’ Theorem; it is necessary to take care
of the possible degeneracies as well.)
The second step involves the use of coordinates to show that, in a Desarguesian
plane, Pappus’ Theorem is equivalent to the commutativity of multiplication. (See
Exercise 3.)
In view of this, projective planes over commutative fields are called Pappian
planes.
Remark. It follows from Theorems 2.1 and 2.3 and Wedderburn’s Theorem 1.1
that, in a finite projective plane, Desargues’ Theorem implies Pappus’. No geometric proof of this implication is known.
A similar treatment of affine planes is possible.
Exercises
1. (a) Show that a collineation which has a centre has an axis, and vice versa.
26
2. Projective planes
'' u
'
'
'
'
'
'
'
'
55 o8 9 9
5 8 8 99
5 3 c8 :! ! ! 9 t
5
3
3 3 5 5 3 b; ; 8; 8 : : 9 9
3
3
8 ; ; :9
3 3 3
5
3
8
3
; 9 a: ;
3
5
3
8
3
9: ; ;
5
3 3
8
3
9 : % % ;s
3
5
8
3p4 34 4
9 7 76 q6 6 6 r
5
8
6
4 4 4 4
7
6
4 4 4 4 5 5 6 6 6 6 8 86 6 7 6 7 6 7 a
4 4 b6 4 6 4
4 4 4 87" 7
c " " "
" " " "
" " "
1
1
1
2
2
2
" ""
v
Figure 2.3: Pappus implies Desargues
(b) Show that a collineation cannot have more than one centre.
2. The group G has a family of proper normal subgroups which partition the
non-identity elements of G. Prove that G is abelian.
>=
>? @=
>? A B ? =
3. In PG 2 F , let the vertices of a hexagon be 1 0 0 , 0 0 1 , 0 1 0 ,
1 α 1 1 , 1 1 0 and β β α 1 1 . Show that alternate vertices lie on the
lines defined by the column vectors 0 0 1 and α 1 1 0 . Show that
opposite sides meet in the points α 0 1 , 0 βα 1 and 1 β α 1 1 . Show
that the second and third of these lie on the line β 1 βα , which also contains
the first if and only if αβ βα.
< 2.3. Projectivities
2.3
27
Projectivities
C
Let Π
X
be a projective plane. Temporarily, let L be the set of points
incident with L; and let x be the set of lines incident with x. If x is not incident
with L, there is a natural bijection between L and x : each point on L lies
on a unique line through x. This bijection is called a perspectivity. By iterating
perspectivities and their inverses, we get a bijection (called a projectivity) between
any two sets x or L . In particular, for any line L, we obtain a set P L of
projectivities from L to itself (or self-projectivities), and analogously a set P x
for any point x.
The sets P L and P x are actually groups of permutations of L or x .
(Any self-projectivity is the composition of a chain of perspectivities; the product
of two self-projectivities corresponds to the concatenation of the chains, while the
inverse corresponds to the chain in reverse order.) Moreover, these permutation
groups are naturally isomorphic: if g is any projectivity from L1 to L2 , say,
then g 1 P L1 g P L2 . So the group P L of self-projectivities on a line is an
invariant of the projective plane. It turns out that the structure of this group carries
information about the plane which is closely related to concepts we have already
seen.
D
# zF
#M F
# F
#z G G F
# GF
# E EE FGu
#
F G
#E E E F G
F G
E# #
E
F
G
E #
F
G
E
M
E
#
F
G
#
F L G
E E
2
2
1
1
x1
x2
y2
y1
Figure 2.4: 3-transitivity
Proposition 2.4 The permutation group P L is 3-transitive.
Proof It suffices to show that there is a projectivity fixing any two points x1 x2
L and mapping any further point y1 to any other point y2 . In general, we will use
28
2. Projective planes
IH
the notation “(L1 to L2 via p)” for the composite of the perspectivities L1
p
and p
L2 . Let Mi be any other lines through xi (i 1 2), u a point on M1 ,
and zi M2 (i 1 2) such that yi uzi are collinear (i 1 2). Then the product of
(L to M1 via z1 ) and (M1 to L via z2 ) is the required projectivity (Fig. 2.4.)
JH
A permutation group G is sharply t-transitive if, given any two t-tuples of
distinct points, there is a unique element of G carrying the first to the second (in
order). The main result about groups of projectivities is the following theorem:
2 2 2- 2- / /
2
2
2
- /
2 ,2 1
2(0 0 2 2 L2 , , 1 1 - - /
( 0 ,
/
( 0 ,0 - - 1 1 /
( , 0 -z
1//
( - 0
p,
1
, -( ( 0 0 0 / / 1 1
, - (
/ 0
1
,(
/
0
L
L
0
-,
( /
0
x. . . . .
. . ( ./ . . .
. .
2
1
3
1
0 101
. . . 0. 1
y
Figure 2.5: Composition of projectivities
Theorem 2.5 The group P L of projectivities on a projective plane Π is sharply
3-transitive if and only if Π is pappian.
Proof We sketch the proof. The crucial step is the equivalence of Pappus’ Theorem to the following assertion:
Let L1 L2 L3 be non-concurrent lines, and x and y two points such
that the projectivity
g
K
(L1 to L2 via x) (L2 to L3 via y)
fixes L1 L3 . Then there is a point z such that the projectivity g is
equal to (L1 to L3 via z).
2.3. Projectivities
29
The hypothesis is equivalent to the assertion that x y and L1 L3 are collinear.
Now the point z is determined, and Pappus’ Theorem is equivalent to the assertion
that it maps a random point p of L1 correctly. (Fig. 2.5 is just Pappus’ Theorem.)
Now this assertion allows long chains of projectivities to be shortened, so that
their action can be controlled.
The converse can be seen another way. By Theorem 2.3, we know that a
Pappian plane is isomorphic to PG 2 F for some commutative field F. Now it is
easily checked that any self-projectivity on a line is induced by a linear fractional
transformation (an element of PGL 2 F ; and this group is sharply 3-transitive.
In the finite case, there are very few 3-transitive groups apart from the symmetric and alternating groups; and, for all known non-Pappian planes, the group of
projectivities is indeed symmetric or alternating (though it is not known whether
this is necessarily so). Both possibilities occur; so, at present, all that this provides
us for non-Pappian finite planes is a single Boolean invariant.
In the infinite case, however, more interesting possibilities arise. If the plane
has order α, then the group of projectivities has α generators, and so has order α;
so it can never be the symmetric group (which has order 2α ). Barlotti  gave an
example in which the stabiliser of any six points is the identity, and the stabiliser
of any five points is a free group. On the other hand, Schleiermacher  showed
that, if the stabiliser of any five points is trivial, then the stabiliser of any three
points is trivial (and the plane is Pappian).
Further developments involve deeper relationships between projectivities, configuration theorems, and central collineations; the definition and study of projectivities in other incidence structures; and so on.
3
Coordinatisation of projective
spaces
In this chapter, we describe axiom systems for projective (and affine) spaces. The
principal results are due to Veblen and Young.
3.1
The GF 2 case
In the last section, we saw an axiomatic characterisation of the geometries
PG 2 F (as projective planes satisfying Desargues’ Theorem). We turn now to
the characterisation of projective spaces of arbitrary dimension, due to Veblen and
Young. Since the points and the subspaces of any fixed dimension determine the
geometry, we expect an axiomatisation in terms of these. Obviously the case of
points and lines will be the simplest.
For the first of several times in these notes, we will give a detailed and selfcontained argument for the case of GF 2 , and treat the general case in rather less
detail.
Theorem 3.1 Let X be a set of points, a set of subsets of X called lines. Assume:
(a) any two points lie on a unique line;
(b) a line meeting two sides of a triangle, not at a vertex, meets the third side;
(c) a line contains exactly three points.
Then X and are the sets of points and lines in a (not necessarily finite dimensional) projective space over GF 2 .
31
32
3. Coordinatisation of projective spaces
htb
x y
x
y
z
y z
x y z x y z
Figure 3.1: Veblen’s Axiom
Remark We will see later that, more or less, conditions (a) and (b) characterise arbitrary projective spaces. Condition (c) obviously specifies that the field
is GF 2 . The phrase “not necessarily finite dimensional” should be interpreted
as meaning that X and can be identified with the subspaces of rank 1 and 2
respectively of a vector space over GF 2 , not necessarily of finite rank.
Proof Since 1 is the only non-zero scalar in GF 2 , the points of projective space
can be identified with the non-zero vectors; lines are then triples of non-zero vectors with sum 0. Our job is to reconstruct this space.
Let 0 be an element not in X , and set V X 0 . Now define an addition in
V as follows:
for all v V , 0 v v 0 v and v v 0;
for all x y X with x y, x y z, where z is the third point of the line
containing x and y.
We claim that V is an abelian group. Commutativity is clear; 0 is the
identity, and each element is its own inverse. Only the associative law is nontrivial; and the only non-trivial case, when x y z are distinct non-collinear points,
follows immediately from Veblen’s axiom (b) (see Fig. 3.1.1).
Next, we define scalar multiplication over GF 2 , in the only possible way:
0 v 0, 1 v v for all v V . The only non-trivial vector space axiom is 1 1 v 1 v 1 v, and this follows from v v 0.
Finally, 0 x y z is a rank 2 subspace if and only if x y z.
There is a different but even simpler characterisation in terms of hyperplanes,
which foreshadows some later developments.
3.1. The GF 2 case
33
Let be any family of subsets of X . The subset Y of X is called a subspace
if any member of which contains two points of Y is wholly contained within Y .
(Thus, the empty set, the whole of X , and any singleton are trivially subspaces.)
The subspace Y is called a hyperplane if it intersects every member of (necessarily in one or all of its points).
Theorem 3.2 Let (a) every set in be a collection of subsets of X . Suppose that
has cardinality 3;
(b) any two points of X lie in at least one member of ;
(c) every point of X lies outside some hyperplane.
Then X and are the point and line sets of a projective geometry over GF 2 , not
necessarily finite dimensional.
Proof Let be the set of hyperplanes. For each point x X , we define a function
px : GF 2 by the rule
px H 01 ifif xx H;
H.
By condition (c), px is non-zero for all x X .
Let P ! px : x X . We claim that P 0 is a subspace of the vector space
GF 2 #" of functions from to GF 2 . Take x y X , and let x y z be any set
in containing x and y. Then a hyperplane contains z if and only if it contains
both or neither of x and y; so pz px py . The claim follows.
Now the map x \$ px is 1–1, since if px py then px py 0, contradicting
the preceding paragraph. Clearly this map takes members of to lines. The
theorem is proved.
Remark The fact that two points lie in a unique line turns out to be a consequence of the other assumptions.
Exercises
1. Suppose that conditions (a) and (c) of Theorem 3.1.2 hold. Prove that two
points of X lie in at most one member of .
34
3. Coordinatisation of projective spaces
2. Let X %& satisfy conditions (a) and (c) of Theorem 3.1.1. Let Y be the
set of points x of X with the following property: for any two lines x y1 y2 and
x z1 z2 containing x, the lines y1 z1 and y2 z2 intersect. Prove that Y is a subspace
of X .
3. Let X be a set of points, ' a collection of subsets of X called lines. Assume
that any two points lie in at least one line, and that every point lies outside some
hyperplane. Show that, if the line size is not restricted to be 3, then we cannot
conclude that X and ' are the point and line sets of a projective space, even if
any two points lie on exactly one line. [Hint: In a projective plane, any line is
a hyperplane. Select three lines L1 L2 L3 forming a triangle. Show that it is
possible to delete some points, and to add some lines, so that L1 L2 L3 remain
hyperplanes.]
4. Let X #'( satisfy the hypotheses of the previous question. Assume additionally that any two points lie on a unique line, and that some hyperplane is a line
and is finite. Prove that there is a number n such that any hyperplane contains n 1
points, any point lies on n 1 lines, and the total number of lines is n2 n 1.
3.2
An application
I now give a brief application to coding theory. This application is a bit spurious, since a more general result can be proved by a different but equally simple argument; but it demonstrates an important link between these fields. Additionally,
the procedure can be reversed, to give characterisations of other combinatorial
designs using theorems about codes.
The problem tackled by the theory of error-correcting codes is to send a message over a noisy channel in which some distortion may occur, so that the errors
can be corrected but the price paid in loss of speed is not too great. This is not the
place to discuss coding theory in detail. We simplify by assuming that a message
transmitted over the channel is a sequence of blocks, each block being an n-tuple
of bits (zeros or ones). We also assume that we can be confident that, during the
transmission of a single block, no more than e bits are transmitted incorrectly (a
zero changed to a one or vice versa). The Hamming distance between two blocks
is the number of coordinates in which they differ; that is, the number of errors required to change one into the other. A code is just a set of “codewords” or blocks
(n-tuples of bits), containing more than one codeword. It is e-error correcting if
the Hamming distance between two codewords is at least 2e 1. (The reason for
the name is that, by the triangle inequality, an arbitrary word cannot lie at distance
3.2. An application
35
e or less from more than one codeword. By our assumption, the received word
lies at distance e or less from the transmitted codeword; so this codeword can be
recovered.)
To maximise the transmission rate, we need as many codewords as possible.
The optimum is obtained when every word lies within distance e of a (unique)
codeword. In other words, the closed balls of radius e centred at the codewords
fill the space of all words without any overlap! A code with this property is called
perfect e-error-correcting.
Encoding and decoding are made much easier if the code is linear, that is, it is
a GF 2 -subspace of the vector space GF 2 n of all words.
Theorem 3.3 A linear perfect 1-error-correcting code has length 2d ) 1 for some
d * 1; there is a unique such code of any length having this form.
Remark These unique codes are called Hamming codes. Their relation to projective spaces will be made clear by the proof below.
Proof Let C be such a code, of length n. Obviously it contains 0. We define the
weight wt v of any word v to be its Hamming distance from 0. The weight of
any non-zero codeword is at least 3. Now let X be the set of coordinate places,
and the set of triples of points of X which support codewords (i.e., for which a
codeword has 1s in just those positions).
We verify the hypotheses of Theorem 3.1. Condition (c) is clear.
Let x and y be coordinate positions, and let w be the word with entries 1
in positions x and y and 0 elsewhere. w is not a codeword, so there must be a
unique codeword c at distance 1 from w; then c must have weight 3 and support
containing x and y. So (a) holds.
Let x y r +, x z q +, y z p be the supports of codewords u v w. By linearity, u v w is a codeword, and its support is p q r . So (b) holds.
Thus X and are the points and lines of a projective space PG d ) 1 2 for
some d * 1; the number of points is n 2d ) 1. Moreover, it’s easy to see that C
is spanned by its words of weight 3 (see Exercise 1), so it is uniquely determined
by d.
Note, incidentally, that the automorphism group of the Hamming code is the
same as that of the projective space, viz. PGL d 2 .
36
3. Coordinatisation of projective spaces
Exercise
1. Prove that a perfect linear code is spanned by its words of minimum weight.
(Use induction on the weight. If w is any non-zero codeword, there is a codeword
u whose support contains e 1 points of the support of w; then u w has smaller
weight than w.)
2. Prove that if a perfect e-error-correcting code of length n exists, then
e
∑
-
i 0
n
i.
is a power of 2. Deduce that, if e 3, then n 7 or 23. (Hint: the cubic polynomial
in n factorises.)
Remark. The case n 7 is trivial. For n 23, there is a unique code (up to
isometry), the so-called binary Golay code.
3. Verify the following decoding scheme for the Hamming code Hd of length
)
d
1. Let Md be the 2d ) 1 / d matrix over GF 2 whose rows are the base
2
2 representations of the integers 1 2 101010# 2d ) 1. Show that the null space of the
matrix Md is precisely Hd . Now let w be received when a codeword is transmitted,
and assume that at most one error has occurred. Prove that
if wHd 0, then w is correct;
if wHd is the ith row of Hd , then the ith position is incorrect.
3.3
The general case
The general coordinatisation theorem is the same as Theorem 3.1, with the
hypothesis “three points per line” weakened to “at least three points per line”.
Accordingly we consider geometries with point set X and line set (where is
a set of subsets of X ) satisfying:
(LS1) Any line contains at least two points.
(LS2) Two points lie in a unique line.
Such a geometry is called a linear space. Recall that a subspace is a set of points
which contains the (unique) line through any two of its points. In a linear space,
in addition to the trivial subspaces (the empty set, singletons, and X ), any line is
3.3. The general case
37
a subspace. Any subspace, equipped with the lines it contains, is a linear space in
its own right.
A linear space is called thick if it satisfies:
(LS1+) Any line contains at least three points.
Finally, we will impose Veblen’s Axiom:
(V) A line meeting two sides of a triangle, not at a vertex, meets the third side
also.
Theorem 3.4 (Veblen–Young Theorem) Let X %& be a linear space, which is
thick and satisfies Veblen’s Axiom (V). Then one of the following holds:
(a) X
/
2! 0;
/
(b) 3 X 34 1, ! 0;
(c)
!5 X , 3 X 376 3;
(d) X %& is a projective plane;
(e) X %& is a projective space over a skew field, not necessarily of finite dimension.
Remark It is common to restrict to finite-dimensional projective spaces by adding
the additional hypothesis that any chain of subspaces has finite length.
Proof (outline)
The key observation provides us with lots of subspaces.
Lemma 3.5 Let X %& be a linear space satisfying Veblen’s axiom. Let Y be a
subspace, and p a point not in Y ; let Z be the union of the lines joining p to points
of Y . Then Z is a subspace, and Y is a hyperplane in Z.
Proof Let q and r be points of Z. There are several cases, of which the generic
case is that where q r Y and the lines pq and pr meet Y in distinct points s, t.
By (V), the lines qr and st meet at a point u of Y . If v is another point of qr, then
by (V) again, the line pv meets st at a point of Y ; so v Z.
38
3. Coordinatisation of projective spaces
We write this subspace as 8 Y p 9 .
Now, if L is a line and p a point not in L, then 8 L p 9 is a projective plane. (It
is a subspace in which L is a hyperplane; all that has to be shown is that every
line is a hyperplane, which follows once we show that 8 L p 9 contains no proper
subspace properly containing a line.)
The theorem is clearly true if there do not exist four non-coplanar points; so
we may suppose that such points do exist.
We claim that Desargues’ Theorem holds. To see this examine the geometric
proof of Desargues’ Theorem in Section 1.2; it is obvious for any non-planar configuration, and the planar case follows by several applications of the non-planar
case. Now the same argument applies here.
It follows from Theorem 2.1 that every plane in our space can be coordinatised
by a skew field.
To complete the proof, we have to show that the coordinatisation can be extended consistently to the whole space. For this, first one shows that the skew
fields coordinatising all planes are the same: this can be proved for planes within
a 3-dimensional subspace by means of central collineations, and the result extends
by connectedness to all pairs of planes. The remainder of the argument involves
careful book-keeping.
From this, we can find a classification of not necessarily thick linear spaces
satisfying Veblen’s axiom. The sum of a family of linear spaces is defined as
follows. The point set is the disjoint union of the point sets of the constituent
spaces. Lines are of two types:
(a) all lines of the constituent spaces;
(b) all pairs of points from different constituents.
It is clearly a linear space.
Theorem 3.6 A linear space satisfying Veblen’s axiom is the sum of linear spaces
of types (b)–(e) in the conclusion of Theorem 3.4.
Proof Let X %& be such a space. Define a relation : on X by the rule that x : y
if either x y, or the line containing x and y is thick (has at least three points).
We claim first that : is an equivalence relation. Reflexivity and symmetry are
clear; so assume that x : y and y : z, where we may assume that x y and z are all
distinct. If these points are collinear, then x : z; so suppose not; let x1 and z1 be
3.4. Lattices
39
further points on the lines xy and yz respectively. By (V), the line x1 z1 contains a
point of xz different from x and z, as required.
So X is the disjoint union of equivalence classes. We show next that any
equivalence class is a subspace. So let x : y. Then x : z for every point z of the
line xy; so this line is contained in the equivalence class of x.
So each equivalence class is a non-empty thick linear space, and hence a point,
line, projective plane, or projective space over a skew field, by Theorem 3.4. It is
clear that the whole space is the sum of its components.
A geometry satisfying the conclusion of Theorem 3.6 is called a generalised
projective space. Its flats are its (linear) subspaces; these are precisely the sums
of flats of the components. The term “projective space” is sometimes extended to
mean “thick generalised projective space” (i.e., to include single points, lines with
at least three points, and not necessarily Desarguesian projective planes).
3.4
Lattices
Another point of view is to regard the flats of a projective space as forming a
lattice. We discuss this in the present section.
A lattice is a set L with two binary operations ; and < (called join and meet),
and two constants 0 and 1, satisfying the following axioms:
(L1) ; and < are idempotent, commutative, and associative;
(L2) x ;= x < y > x and x <= x ; y x;
(L3) x < 0 x, x ; 1 x.
It follows from these axioms that x < y x holds if and only if x ; y y holds.
We write x ? y if these equivalent conditions hold. Then L 1?( is a partially
ordered set with greatest element 1 and least element 0; x ; y and x < y are the
least upper bound and greatest lower bound of x and y respectively. Conversely,
any partially ordered set in which least upper bounds and greatest lower bounds
of all pairs of elements exist, and there is a least element and a greatest element,
gives rise to a lattice.
In a lattice, an atom is a non-zero element a such that a < x 0 or a for any
x; in other words, an element greater than zero but minimal subject to this. The
lattice is called atomic if every element is a join of atoms.
A lattice is modular if it satisfies:
40
3. Coordinatisation of projective spaces
(M) If x ? z, then x ;= y < z @ x ; y A< z for all y.
(Note that, if x ? z, then x ;= y < z B?C x ; y D< z in any lattice.)
Theorem 3.7 A lattice is a generalised projective space of finite dimension if and
only if it is atomic and modular.
Proof The forward implication is an exercise. Suppose that the lattice L is atomic
and modular. Let X be the set of atoms. Identify every element z of the lattice with
the set x X : x ? z . (This map is 1–1; it translates meets to intersections, and
the lattice order to the inclusion order.)
Let x y z be atoms, and suppose that z ? x ; y. Then trivially x ; z ? x ; y.
Suppose that these two elements are unequal. Then y ? x ; z. Since y is an atom,
y <& x ; z 0, and so x ;& y <& x ; z 1 x. But x ; y E<& x ; z F x ; z, contradicting
modularity. So x ; z x ; y. Hence, if we define lines to be joins of pairs of atoms,
it follows that two points lie in a unique line.
Now we demonstrate Veblen’s axiom. Let u v be points on x ; y, x ; z respectively, where xyz is a triangle. Suppose that y ; z G<H u ; v F 0. Then y ; u ; v 6 z,
so y ; u ; v 6 y ; z; in other words, y ;I u ; v JK< y ; z L y ; z. On the other hand,
y ;=J u ; v D<= y ; z J y ; 0 y, contradicting modularity. So the lines y ; z and
u ; v meet.
By Theorem 3.6, the linear subspace is a generalised projective geometry.
Clearly the geometry has finite dimension. We leave it as an exercise to show
that every flat of the geometry is an element of the lattice.
Exercises
1. Complete the proof of Theorem 3.7.
2. Show that an atomic lattice satisfying the distributive laws is modular, and
deduce that it is isomorphic to the lattice of subsets of a finite set.
3.5
Affine spaces
Veblen’s axiom in a linear space is equivalent to the assertion that three noncollinear points lie in a subspace which is a projective plane. It might be hoped that
replacing “projective plane” by “affine plane” here would give an axiomatisation
of affine spaces. We will see that this is almost true.
3.5. Affine spaces
41
Recall from Section 2.1 the definition of an affine plane, and the fact that parallelism is an equivalence relation in an affine plane, where two lines are parallel
if they are equal or disjoint.
Now suppose that X %M is a linear space satisfying the following condition:
(AS1) There is a collection N of subspaces with the properties that each member
of N is an affine plane, and that any three non-collinear points are contained
in a unique member of N .
First, a few remarks about such spaces.
1. All lines have the same cardinality. For two intersecting lines lie in an affine
plane, and so are equicardinal; and, given two disjoint lines, there is a line meeting
both.
2. It would be simpler to say “any three points generate an affine plane”, where
the subspace generated by a set is the intersection of all subspaces containing it.
This formulation is equivalent if the cardinality of a line is not 2. (Affine spaces
of order greater than 2 have no non-trivial proper subspaces.) But, if lines have
cardinality 2, then any pair of points is a line, and so any three points form a subspace which is a generalised projective plane. However, we do want a formulation
which includes this case.
3. In a linear space satisfying (AS1), two lines are said to be parallel if either
they are equal, or they are disjoint and contained in a member of N (and hence
parallel there). Now Playfair’s Axiom holds: given a line L and point p, there is a
unique line parallel to L and containing p. Moreover, parallelism is reflexive and
symmetric, but not necessarily transitive. We will impose the further condition:
(AS2) Parallelism is transitive.
Theorem 3.8 A linear space satisfying (AS1) and (AS2) is empty, a single point,
a single line, an affine plane, or the configuration of points and lines in a (not
necessarily finite-dimensional) affine space.
Proof Let X %& be the linear space. We may assume that it is not empty, a
point, a line, or an affine plane (i.e., that there exist four non-coplanar points).
Step 1. Define a solid to be the union of all the lines in a parallel class C which
meet a plane Π &N , where Π contains no line of C. Then any four non-coplanar
points lie in a unique solid, and any solid is a subspace.
42
3. Coordinatisation of projective spaces
That a solid is a subspace is shown by considering cases, of which the generic
one runs as follows. Let p q be points such that the lines of C containing p and q
meet Π in distinct points x and y. Then x y p q lie in an affine plane; so the line
of C through a point r of pq meets Π in a point x of xy.
Now the fact that the solid is determined by any four non-coplanar points
follows by showing that it has no non-trivial proper subspaces except planes (if
the cardinality of a line is not 2) or by counting (otherwise).
In a solid, if a plane Π contains no parallel to a line L, then Π meets L in a
single point. Hence any two planes in a solid are disjoint or meet in a line.
Step 2. If two planes Π and ΠO contain lines from two different parallel classes,
then every line of Π is parallel to a line of ΠO .
Suppose not, and let L M N be lines of Π, concurrent at p, and pO a point of
ΠO such that the lines LO M O through pO parallel to L and M lie in ΠO , but the line
N O parallel to N does not. The whole configuration lies in a solid; so the planes
NN O and ΠO , with a common point pO , meet in a line K. Now K is coplanar with N
but not parallel to it, so K P N is a point q. Then Π and ΠO meet in q, and hence in
a line J. But then J is parallel to both L and M, a contradiction.
We call two such planes parallel.
Step 3. We build the embedding projective space. Here I will use a typographic
convention to distinguish the two related spaces: elements of the space we are
building will be written in CAPITALS. The POINTS are the points of X and the
parallel classes of lines of N . The LINES are the lines of and the parallel classes
of planes in N . Incidence is hopefully obvious: as in the old space, together with
incidence between any line and its parallel class, as well as between a parallel
class C of lines and a parallel class Q of planes if a plane in Q contains a line in C.
By Step 2, this is a linear space; and clearly every LINE contains at least three
POINTS. We call the new POINTS and LINES (i.e., the parallel classes) “ideal”.
Step 4. We verify Veblen’s Axiom. Any three points which are not all “ideal” lie
in an affine plane with its points at infinity adjoined, i.e., a projective plane. So let
pqr be a triangle of “ideal” POINTS, s and t POINTS on pq and pr respectively,
and o a point of X . Let P Q R S T be the lines through o in the parallel classes
p q r s t respectively. Then these five lines lie in a solid, so the planes QR and
ST (having the point o in common) meet in a line u. The parallel class U of u is
the required POINT on qr and st.
3.6. Transitivity of parallelism
43
By Theorem 3.4, the extended geometry is a projective space. The points at
infinity obviously form a hyperplane, and so the original points and lines form an
affine space.
We spell the result out in the case where lines have cardinality 2, but referring
only to parallelism, not to the planes.
Corollary 3.9 Suppose that the 2-element subsets of a set X are partitioned into
“parallel classes” so that each class partitions X . Suppose that, for any four
points p q r s X , if pq R rs, then pr R qs. Then the points and parallelism are
those of an affine space over GF 2 .
Here, we have used the notation R to mean “belong to the same parallel class
as”. The result follows immediately from the theorem, on defining N to be the set
of 4-element subsets which are the union of two parallel 2-subsets.
Exercises
1. Give a direct proof of the Corollary, in the spirit of Section 3.1.
3.6
Transitivity of parallelism
A remarkable theorem of Buekenhout  shows that it is not necessary to
assume axiom (AS2) (the transitivity of parallelism) in Theorem 3.8, provided
that the cardinality of a line is at least 4. Examples due to Hall  show that the
condition really is needed if lines have cardinality 3.
Theorem 3.10 Let X %& be a linear space satisfying (AS1), in which some line
contains at least four points. Then parallelism is transitive (that is, (AS2) holds),
and so X %M is an affine space.
To discuss the counterexamples with 2 or 3 points on a line, some terminology
is helpful. A Steiner triple system is a collection of 3-subsets of a set, any two
points lying in a unique subset of the collection. In other words, it is a linear
space with all lines of cardinality 3, or (in the terminology of Section 1.4) a 2 v 3 1 design for some (possibly infinite) v. A Steiner quadruple system is a set
of 4-subsets of a set, any three points in a unique subset in the collection (that is,
a 3- v 4 1 design.)
44
3. Coordinatisation of projective spaces
A linear space satisfying (AS1), with two points per line, is equivalent to a
Steiner quadruple system: the distinguished 4-sets are the affine planes. There
are Steiner quadruple systems aplenty; most are not affine spaces over GF 2 (for
example, because the number of points is not a power of 2). Here is an example.
Let A C 1 2 3 4 5 6 . Let X be the set of all partitions of A into two sets of size
3 (so that 3 X 3S 10). Define two types of 4-subsets of X :
(a) for all a b A, the set of partitions for which a b lie in the same part;
(b) for all partitions of A into three 2-sets A1 A2 A3 , the set of all partitions into
two 3-sets each of which is a transversal to the three sets Ai .
This is a Steiner quadruple system with 10 points.
In the case of three points per line, we have the following result, for which we
refer to Bruck [D] and Hall [18, 19]:
Theorem 3.11 (a) In a finite Steiner triple system satisfying (AS1), the number
of points is a power of 3.
(b) For every d 6 4, there is a Steiner triple system with 3d points which is not
isomorphic to AG d 3 .
Exercises
1. Prove that the number of points in a Steiner triple system is either 0 or
congruent to 1 or 3 (mod 6), while the number of points in a Steiner quadruple
system is 0, 1, or congruent to 2 or 4 (mod 6).
(It is known that these conditions are sufficient for the existence of Steiner
triple and quadruple systems.)
2. Let X %& be a Steiner triple system satisfying (AS1). For each point x X ,
let τx be the permutation of X which fixes x and interchanges y and z whenever
x y z is a triple. Prove that
(a) τx is an automorphism;
(b) τ2x 1;
(c) for x y, τx τy 3
1.
4
Various topics
This chapter collects some topics, any of which could be expanded into an entire
chapter (or even a book!): spreads and translation planes; subsets of projective
spaces; projective lines; and the simplicity of PSL n F .
4.1
Spreads and translation planes
Let V be a vector space over F, having even rank 2n. A spread is a set of
subspaces of V of rank n, having the property that any non-zero vector of V lies
in a unique member of . A trivial example occurs when n 1 and consists of
all the rank 1 subspaces.
The importance of spreads comes from the following result, whose proof is
straightforward.
Proposition 4.1 Let be a spread in V , and the set of all cosets of members of
. Then V is an affine plane. The projective plane obtained by adding a line
at infinity L∞ is p L∞ -transitive for all p L∞ .
For finite planes, the converse of the last statement is also true. An affine plane
with the property that the projective completion is p L∞ -transitive for all p L∞
is called a translation plane.
Example. Let K be an extension field of F with degree n. Take V to be a rank
2 vector space over K, and the set of rank 1 K-subspaces. Then, of course,
the resulting affine plane is AG 2 K . Now forget the K-structure, and regard V
45
46
4. Various topics
as an F- vector space. Such a spread is called Desarguesian, because it can be
recognised by the fact that the affine plane is Desarguesian.
Projectively, a spread is a set of n 1 -dimensional flats in PG 2n 1 F ,
which partitions the points of F. We will examine further the case n 1, which
will be considered again in section 4.5. Assume that F is commutative.
Lemma 4.2 Given three pairwise skew lines in PG 3 F , there is a unique common transversal through any point on one of the lines.
Proof Let L1 L2 L3 be the lines, and p L1 . The quotient space by p is a projective plane PG 2 F , and Π1 p L2 and Π2 p L3 are distinct lines in this
plane; they meet in a unique point, which corresponds to a line M containing p
and lying in Π1 and Π2 , hence meeting L2 and L3 .
Now let be the set of common transversals to the three pairwise skew lines.
The lines in are pairwise skew, by 4.2.
Lemma 4.3 A common transversal to three lines of is a transversal to all of
them.
For the proof, see Exercise 2, or Section 8.4.
Let be the set of all common transversals to . The set is called a
regulus, and (which is also a regulus) is the opposite regulus. Thus, three
pairwise skew lines lie in a unique regulus.
A spread is regular if it contains the regulus through any three of its lines.
Theorem 4.4 A spread is Desarguesian if and only if it is regular.
(The proof of the forward implication is given in Exercise 2.)
If we take a regular spread, and replace the lines in a regulus in this spread
by those in the opposite regulus, the result is still a spread; for a regulus and its
opposite cover the same set of points. This process is referred to as derivation. It
gives rise to non-Desarguesian translation planes:
Proposition 4.5 If F 2, then a derivation of a regular spread is not regular.
Proof Choose two reguli 1 , 2 with a unique line in common. If we replace 1
by its opposite, then the regulus 2 contains three lines of the spread but is not
contained in the spread.
4.1. Spreads and translation planes
47
It is possible to push this much further. For example, any set of pairwise disjoint reguli can be replaced by their opposites. I will not discuss this any further.
The concept of a spread of lines in PG 3 F can be dualised. (For the rest
of the section, F is not assumed commutative.) A set of pairwise skew lines is
called a cospread if every plane contains a (unique) line of ; in other words, if corresponds to a spread in the dual space PG 2 F . Call a bispread if it is both
a spread and a cospread.
If F is finite, then every spread is a bispread. (For there are equally many, viz.
q 1 q2 1 , points and planes; and a set of n pairwise skew lines accounts for
q 1 n points and the same number of planes.) Moreover, a Desarguesian spread
is a bispread; and any derivation of a bispread is a bispread (since the concept of a
regulus is self-dual). The reader may be wondering if there are any spreads which
are not bispreads! That they exist in profusion is a consequence of the next result
/ and gives us lots of strange translation planes.
(take 0),
Theorem 4.6 Let F be an infinite field. Let , be sets of points and planes in
PG 3 F , with the property that F . Then there is a set of pairwise
skew lines, satisfying
(a) the point p lies on a line of if and only if p "
! ;
! .
(b) the plane Π contains a line of if and only if Π #
Proof We use the fact that PG 2 F is not the union of fewer than F points and
lines. For, if S is any set of fewer than F points and lines, and L is a line not in
S, then L is not covered by its intersections with members of S.
The proof is a simple transfinite induction. (Note that we are using the Axiom
of Choice here; but, in any case, the proof is valid over any field which can be
well-ordered, in particular, over any countable field.) For readers unfamiliar with
set theory, assume that F is countable, delete the word “transfinite”, and ignore
comments about limit ordinals in the following argument.
Let α be the initial ordinal of cardinality F . Well-order the points of PG 3 F not in and the planes not in in a single sequence of order-type α, say Xβ :
β α . Construct a sequence β : β α by transfinite recursion, as follows.
/
Set 0 0.
Suppose that β is a successor ordinal, say β γ 1. Suppose that Xβ is a point
(the other case is dual). If γ contains a line incident with Xβ, then set β \$ γ .
Suppose not. Consider the projective plane PG 3 F &% Xβ. By our initial remark,
48
4. Various topics
this plane is not covered by fewer than α lines of the form L Xβ % Xβ (for L ' γ )
or Π % Xβ (for Π ( with Xβ Π) and points p Xβ % Xβ (for p " ). So we can
choose a point lying outside the union of these points and lines, that is, a line Lβ
containing Xβ so that Lβ ) L 0/ (for L * γ ), Lβ ! Π (for Π + ), and p ! Lβ (for
p " ). Set β ( γ ,.- Lβ / .
If β is a limit ordinal, set
Then α
γ3
β 10
γ2 β
is the required set of lines.
Exercises
1. Show that, if three pairwise skew lines in PG 3 F are given, then it is
possible to choose coordinates so that the lines have equations
x1 x2
0;
x3 x4
0;
x3 x1 x4
x2 .
Find the common transversals to these three lines.
2. Now let F be commutative. Show that the common transversals to any three
of the lines found in the last question are the original three lines and the lines with
equations
x1 x3 α, x2
x4 α
for α F, α ! 0 1.
Deduce that the Desarguesian spread defined by a quadratic extension of F is
regular.
3. Prove that Lemma 4.3 is valid in PG 3 F if and only if F is commutative.
4. Use Theorem 4.6 to show that, if F is an infinite field, then there is a spread
of lines in AG 3 F which contains one line from each parallel class.
4.2
Some subsets of projective spaces
For most of the second half of these lecture notes, we will be considering subsets of projective spaces which consist of the points (and general subspaces) on
4.2. Some subsets of projective spaces
49
which certain forms vanish identically. In this section, I will describe some more
basic subsets of projective spaces, and how to recognise them by their intersections with lines. The first example is a fact we have already met.
Proposition 4.7 (a) A set S of points in a projective space is a subspace if and
only if, for any line L, S contains no point, one point, or all points of L.
(b) A set S of points in a projective space is a hyperplane if and only if, for any
line L, S contains one or all points of L.
The main theorem of this section is a generalisation of Proposition 4.7(a).
What if we make the condition symmetric, that is, ask that S contains none, one,
all but one, or all points of any line L? The result is easiest to state in the finite
case:
Theorem 4.8 Let S be a set of points of X PG n F such that, for any line L,
S contains none, one, all but one, or all points of S. Suppose that F 4 2. Then
there is a chain
0/ X0 5 X1 563&3&375 Xm X
of subspaces of X , such that either S
X2i ; 1 .
98
i: 0
X2i ;
1<
X2i , or S
98
i: 0
X2i ;
2<
The hypothesis that F 4 2 is necessary: over the field GF 2 , a line has just
three points, so the four possibilities listed in the hypothesis cover all subsets of
a line. This means that any subset of the projective space satisfies the hypothesis!
(Nevertheless, see Theorem 4.10 below.)
Note that the hypothesis on S is “self-complementary”, and the conclusion
must reflect this. It is more natural to talk about a colouring of the points with two
colours such that each colour class satisfies the hypothesis of the theorem. In this
language, the result can be stated as follows.
Theorem 4.9 Let the points of a (possibly infinite) projective space X over F be
coloured with two colours c1 and c2 , such that every colour class contains none,
one, all but one, or all points of any line. Suppose that F 7 2. Then there is a
chain = of subspaces of X , and a function f : =#> - c1 c2 / , so that
(a) 8
=?
X;
(b) for Y @= , there exist points of Y lying in no smaller subspace in = , and all
such points have colour f Y .
The proof proceeds in a number of stages.
50
4. Various topics
Step 1 The result is true for a projective plane (Exercise 1).
Now we define four relations
B
,
1, 2, A
on X , as follows:
B
p 1 q if p is the only point of its colour on the line pq; (this relation or
its converse holds between p and q if and only if p and q have different
colours);
B
p 2 q if there exists q with p
have the same colour);
B
p
q if p
1
q or p
p A q if neither p
colour).
2
1
r and r
1
q (this holds only if p and q
q;
q nor q
p (this holds only if p and q have the same
Step 2 There do not exist points p q with p 2 q and q 2 p.
For, if so, then (with p1 p, q1 q) there are points p2 q2 such that
p1
1
p2
1
q1
1
q2
2
p1 3
Let ci be the colour of pi and qi , i 1 2. By Step 1, the colouring of the plane
p1 p2 q1 is determined; and every point of this plane off the line p1 p2 . In particular, if x1 p1 q1 , x1 ! p1 q1 , then every point of x1 p2 except p2 has colour c1 .
Similarly, every point of x1 q2 except q2 has colour c1 ; and then every point of x1 x2
except x2 has colour c1 , where x2 p2 q2 , x2 ! p2 q2 .
But, by the same argument, every point of x1 x2 except x1 has colour c2 , giving
a contradiction.
Step 3 is a partial order.
The antisymmetry follows by definition for 1 and by Step 2 for 2 ; we must
prove transitivity. So suppose that p q r, and consider cases. If p 1 q 1 r,
then p 2 r by definition. If p 1 q 2 r or p 2 q 1 r, then p and r have
different colours and so are comparable; and r 1 p contradicts Step 2. Finally, if
p 2 q 2 r, then p 1 s 1 q for some s; then s 1 r, so that p 2 r.
Step 4 If p q A r or p A q r, then p r; and if p A q A r, then p A r.
Suppose that p q A r. If p 1 q, then p and r have different colours and so are
comparable; and r 1 p would imply r q by Step 3, so p 1 r. The next case is
similar. The last assertion is a simple consequence of the other two.
4.3. Segre’s Theorem
51
Step 5 If p q, then p r for all points r of pq except p; and the points of pq
other than p are pairwise incomparable.
This holds by assumption if p 1 q, and by the proof of Step 2 if p 2 q.
Now let S p C
-
q : p ! q / , and T p D
-
q : q p/ .
Step 6 S p and T p are subspaces, with p S p < T p . Moreover, T p is
the union of the spaces S q for q p, and is spanned by the points of S p with
colour different from that of p; and we have
p A q implies S p C S q ;
q p implies S q FE T p .
All of this follows by straightforward argument from the preceding steps.
Now the proof of the Theorem follows: we set =G - S p : p X / , and let
f S p & be the colour of p. The conclusions of the Theorem follow from the
assertions in Step 6.
Remark. The only place in the above argument where the hypothesis F H 2
was used was in Step 1. Now PG 2 2 has seven points; so, up to complementation, a subset of PG 2 2 is empty, a point, a line with a point removed, a line, or
a triangle. Only the last case fails to satisfy the conclusion of the Theorem. So we
have the following result:
Theorem 4.10 The conclusions of Theorems 4.8 and 4.9 remain true in the case
F GF 2 provided that we add the extra hypothesis that no colour class intersects a plane in a triangle (or, in 4.8, that no plane meets S in a triangle or the
complement of one).
Exercise
1. Prove that Theorem 4.8 holds in any projective plane of order greater than 2
(not necessarily Desarguesian).
4.3
Segre’s Theorem
For projective geometries over finite fields, it is very natural to ask for characterisations of interesting sets of points by hypotheses on their intersections with
52
4. Various topics
lines. Very much finer discriminations are possible with finite than with infinite
cardinal numbers; for example, all infinite subsets of a countably infinite set whose
complements are also infinite are alike.
It is not my intention to survey even a small part of this vast literature. But I
will describe one of the earliest and most celebrated results of this kind. I begin
with some generalities about algebraic curves. Assume that F is a commutative
field.
If a polynomial f in x1 3&3I3 xn ; 1 is homogeneous, that is, a sum of terms all of
the same degree, then f v C 0 implies f αv C 0 for all α F. So, it f vanishes
at a non-zero vector, then it vanishes at the rank 1 subspace (the point of PG n F )
it spans. The algebraic variety defined by f is the set of points spanned by zeros
of f . We are concerned here only with the case n 2, in which case (assuming
that f does not vanish identically) this set is called an algebraic curve.
Now consider the case where f has degree 2, and F GF q , where q is an
odd prime power. The curve it defines may be a single point, or a line, or two lines;
but, if none of these occurs, then it is equivalent (under the group PGL 3 q ) to
the curve defined by the equation x21 x22 x23 0 (see Exercise 1). Any curve
equivalent to this one is called a conic (or irreducible conic).
It can be shown (see Exercise 2) that a conic has q 1 points, no three of
which are collinear. The converse assertion is the content of Segre’s Theorem:
Theorem 4.11 (Segre’s Theorem) For q odd, a set of q 1 points in PG 2 q ,
with no three collinear, is a conic.
Proof Let J be an oval. We begin with some combinatorial analysis which applies in any plane of odd order; then we introduce coordinates.
Step 1 Any point not on
J
lies on 0 or 2 tangents.
Proof Let p be a point not on J . Since KJLM q 1 is even, and an even number
of points lie on secants through p, an even number must lie on tangents also. Let
xi be the number of points outside J which lie on i tangents. Now we have
∑i i ∑ xi
∑ ixi
1 xi
q2 q 1 q q 1 q 3
(These are all obtained by double counting. The first holds because there are q2
points outside J ; the second because there are q 1 tangents (one at each point
4.3. Segre’s Theorem
53
of J ), each containing q points not on J ; and the third because any two tangents
intersect at a unique point outside J .)
From these equations, we see that ∑ i i 2 xi 0. But the term i 1 in the
sum vanishes (any point lies on an even number of tangents); the terms i 0 and
i 2 clearly vanish, and i i 2 D 0 for any other value of i. So xi 0 for all i ! 0
or 2, proving the assertion.
Remark Points not on J are called exterior points or interior points according
as they lie on 2 or 0 tangents, by analogy with the real case. But the analogy goes
no further. In the real case, every line through an interior point is a secant; this is
false for finite planes.
Step 2
The product of all the non-zero elements of GF q is equal to
1.
Proof The solutions of the quadratic x2 1 are x 1 and x 1; these are the
only elements equal to their multiplicative inverses. So, in the product of all the
non-zero elements, everything except 1 and 1 pairs off with its inverse, leaving
these two elements unpaired.
For the next two steps, note that we can choose the coordinate system so that
the sides of a given triangle have equations x 0, y 0 and z 0 (and the opposite
vertices are N 1 0 0O , N 0 1 0O , and N 0 0 1O respectively). We’ll call this the triangle
of reference.
Step 3 Suppose that concurrent lines through the vertices of the triangle of reference meet the opposite sides in the points N 0 1 aO , N b 0 1O , and N 1 c 0O . Then
abc 1.
Proof The equations of the concurrent lines are z ay, x bz and y cx respectively; the point of concurrency must satisfy all three equations, whence abc 1.)
Remark This result is equivalent to the classical Theorem of Menelaus.
Step 4 Let the vertices of the triangle of reference be chosen to be three points
of J , and let the tangents at these points have equations z ay, x bz and y cx
respectively. Then abc 1.
54
4. Various topics
Proof There are q 2 further points of J , say p1 3&3&3 pq P 2 . Consider the point
N 1 0 0O . It lies on the tangent z ay, meeting the opposite side in N 0 1 aO ; two secants which are sides of the triangle; and q 2 further secants, through p1 3&3&3 pq P 2 .
qP 2
Let the secant through pi meet the opposite side in N 0 1 ai O . Then a ∏i Q 1 ai 6 1,
qP 2
qP 2
by Step 2. If bi ci are similarly defined, we have also b ∏i Q 1 bi c ∏i Q 1 ci R 1.
Thus
qP 2
abc ∏ ai bi ci C 1 3
iQ 1
But, by Step 3, ai bi ci
1 for i 1 3&3I3 q 2; so abc 1.
Step 5 Given any three points p q r of J , there is a conic
p q r and having the same tangents at these points as does J .
=
passing through
Proof Choosing coordinates as in Step 4, the conic with equation
yz czx caxy 0
can be checked to have the required property. (For example, N 1 0 0O lies on this
conic; and, putting z ay, we obtain ay2 0, so N 1 0 0O is the unique point of the
conic on this line.)
Step 6 Now we are finished if we can show that the conic
through an arbitrary further point s of J .
=
of Step 5 passes
Proof Let = and = be the conics passing through p q s and p r s respectively
and having the correct tangents there. Let the conics = , = and = S have equations
f 0, f 0, f 0 respectively. (These equations are determined up to a
constant factor.) Let L p Lq Lr Ls be the tangents to J at p q r s respectively.
Since all three conics are tangent to L p at p, we can choose the normalisation so
that f f f S agree identically on L p .
Now consider the restrictions of f and f to Ls . Both are quadratic functions
having a double zero at s, and the values at the point Ls ) L p coincide; so the
two functions agree identically on Ls . Similarly, f and f agree on Lq , and f
and f agree on Lr . But then f , f and f all agree at the point Lq ) Lr . So the
quadratic functions f and f agree on L p , Ls , and Lq ) Lr , which forces them to
be equal. So the three conics coincide, and our claim is proved (and with it Segre’s
Theorem).
4.3. Segre’s Theorem
55
The argument in the last part of the proof can be generalised to give the following result (of which it forms the case n q 1, m 2, with L1 3&3&3 Ln the tangents
to the oval, and - pi1 pi2 / the point of tangency of Li taken with multiplicity 2).
Proposition 4.12 Let L1 3&3&3 Ln be lines in PG 2 q , no three concurrent. Let
pi1 3&3&3 pim be points of Li , not necessarily distinct, but lying on none of the other
LiT . Suppose that, for any three of the lines, there is an algebraic curve of degree
m whose intersections with those lines are precisely the specified points (counted
with the appropriate multiplicity). Then there is a curve of degree m, meeting each
line in just the specified points.
Proposition 4.12 has been generalised  to arbitrary sets of lines (without
the assumption that no three are concurrent).
Proposition 4.13 Let L1 3&3I3 Ln be lines in PG 2 q . Let pi1 3&3&3 pim be points of
Li , not necessarily distinct, but lying on none of the other LiT . Suppose that, for any
three of the lines which form a triangle, and for the set of all lines passing through
any point of the plane (whenever there are at least three such lines), there is an
algebraic curve of degree m whose intersections with those lines are precisely the
specified points (counted with the appropriate multiplicity). Then there is a curve
of degree m, meeting each line in just the specified points.
The analogue of Segre’s Theorem over GF q with even q is false. In this case,
the tangents to an oval S all pass through a single point n, the nucleus of the oval
(Exercise 4); and, for any p S, the set S ,U- n / <V- p / is also an oval. But, if q 4,
then at most one of these ovals can be a conic (see Exercise 5: these ovals have
q common points). For sufficiently large q (viz., q W 64), and also for q 16,
there are other ovals, not arising from this construction. We refer to  or  for
up-to-date information on ovals in planes of even order.
We saw that there are ovals in infinite projective planes which are not conics.
However, there is a remarkable characterisation of conics due to Buekenhout. A
hexagon is said to be Pascalian if the three points of intersection of opposite sides
are collinear. In this terminology, Pappus’ Theorem asserts that a hexagon whose
vertices lie alternately on two lines is Pascalian. Since a pair of lines forms a
“degenerate conic”, this theorem is generalised by Pascal’s Theorem:
Theorem 4.14 (Pascal’s Theorem) In a Pappian projective plane, a hexagon inscribed in a conic is Pascalian.
56
4. Various topics
We know from Theorem 2.3 that a projective plane satisfying Pappus’ Theorem is isomorphic to PG 2 F for a commutative field F. The theorem of Buekenhout completes this circle of ideas. Its proof is group-theoretic, using a characterisation of PGL 2 F as sharply 3-transitive group due to Tits.
Theorem 4.15 (Buekenhout’s Theorem) Let S be an oval in a projective plane
Π. Suppose that every hexagon with vertices in S is Pascalian. Then Π is isomorphic to PG 2 F for some commutative field F, and S is a conic in Π.
Exercises
1. (a) By completing the square, prove that any homogeneous polynomial of
degree 2 in n variables, over a commutative field F with characteristic different
from 2, is equivalent (by non-singular linear transformation) to the polynomial
α1 x21 αn x2n 3
3&3&3 (b) Prove that multiplication of any αi in the above form by a square in F gives
an equivalent form.
(c) Now let F GF q and n 3; let η be a fixed nonsquare in F. Show that
the curves defined by x21 , x21 x22 and x21 ηx22 are respectively a line, two lines,
and a point. Show that there exists α such that η 1 α2 . Observing that
x αy 2
αx y 2
η x2 y2 M
prove that the forms x21 x22 x23 and x21 ηx22 ηx23 are equivalent. Deduce the
classification of curves of degree 2 over GF q given in the text.
2. Count the number of secants through an exterior point and through an
interior point of an oval in a projective plane of odd order q. Also, count the
number of points of each type.
3. Prove that a curve of degree 2 over any commutative field is empty, a point,
a line, a pair of lines, or an oval. Prove also that a curve of degree 2 over a finite
field is non-empty.
4. Prove that, if q is even, then the tangents to an oval in a projective plane
of order q are concurrent. Deduce that there is a set of q 2 points with no three
collinear, having no tangents (i.e., meeting every line in 0 or 2 points). (Removing
any one of these points then gives an oval.)
4.4. Ovoids and inversive planes
57
Remark. A set of n 2 points in a plane of order n, no three collinear, is called
a hyperoval.
5. Prove that, in any infinite projective plane, for any integer k 1, there is a
set of points meeting every line in exactly k points.
6. Prove that five points of PG 2 F , with no three collinear, are contained in
a unique conic. (Take four of the points to be the standard set 1 0 0 , 0 1 0 ,
0 0 1 and 1 1 1 ; the fifth is 1 α β , where α and β are distinct from one
another and from 0 and 1.)
4.4
Ovoids and inversive planes
Ovoids are 3-dimensional analogues of ovals. They have added importance
because of their connection with inversive planes, which are one-point extensions
of affine planes. (The traditional example is the relation between the Riemann
sphere and the “extended complex plane”.)
Fields in this section are commutative.
An ovoid in PG 3 F is a set J of points with the properties
(O1) no three points of
(O2) the tangents to
J
J
are collinear;
through a point of
J
form a plane pencil.
(If a set of points satisfies (O1), a line is called a secant, tangent or passant if it
meets the set in 2, 1 or 0 points respectively. The plane containing the tangents to
an ovoid at a point x is called the tangent plane at x.)
The classical examples of ovoids are the elliptic quadrics. Let αx2 βx γ
be an irreducible quadratic over the field F. The elliptic quadric consists of the
points of PG 3 F whose coordinates x1 x2 x3 x4 satisfy
x1 x2 αx23 βx3 x4 γx24
03
The proof that these points do form an ovoid is left as an exercise.
Over finite fields, ovoids are rare. Barlotti and Panella showed the following
analogue of Segre’s theorem on ovals:
Theorem 4.16 Any ovoid in PG 3 q , for q an odd prime power, is an elliptic
quadric.
58
4. Various topics
For even q, just one further family is known, the Suzuki–Tits ovoids, which we
will construct in Section 8.4.
An inversive plane is, as said above, a one-point extension of an affine plane.
That is, it is a pair X X=Y , where X is a set of points, and = a collection of subsets
of X called circles, satisfying
(I1) any three points lie in a unique circle;
(I2) if x y are points and C a circle with x
circle C satisfying y C and C ) C 7
C and y
- x/ ;
%
C, then there is a unique
(I3) there exist four non-concircular points.
It is readily checked that, for x X , the points different from x and circles containing x form an affine plane. The order of the inversive plane is the (common)
order of its derived affine planes.
Proposition 4.17 The points and non-trivial plane sections of an ovoid form an
inversive plane.
Proof A plane section of the ovoid J is non-trivial if it contains more than one
point. Any three points of J are non-collinear, and so define a unique plane
section. Given x, the points of J different from x and the circles containing x
correspond to the lines through x not in the tangent plane Tx and the planes through
x different from Tx ; these are the points of the quotient space not incident with the
line Tx % x and the lines different from Tx % x, which form an affine plane.
An inversive plane arising from an ovoid in this way is called egglike. Dembowski proved:
Theorem 4.18 Any inversive plane of even order is egglike (and so its order is a
power of 2).
This is not known to hold for odd order, but no counterexamples are known.
There are configuration theorems (the bundle theorem and Miquel’s theorem
respectively) which characterise egglike inversive planes and “classical” inversive
planes (coming from the elliptic quadric) respectively.
Higher-dimensional objects can also be defined. A set J of points of PG n F is an ovoid if
(O1) no three points of
J
are collinear;
4.5. Projective lines
(O2’) the tangents to J through a point x of
hyperplane of PG n F .
59
are all the lines through x in a
J
Proposition 4.19 If F is finite and n W 4, then PG n F contains no ovoid.
However, there can exist such ovoids over infinite fields (Exercise 3).
Exercises
1. Prove Proposition 4.19. [Hint: it suffices to prove it for n 4.]
2. Prove that, for q odd, a set of points in PG 3 q which satisfies (O1) has
cardinality at most q2 1, with equality if and only if it is an ovoid.
(This is true for q even, q 2 also, though the proof is much harder. For q 2,
the complement of a hyperplane is a set of 8 points in PG 3 2 satisfying (O1).)
3. Show that the set of points of PG n [Z\ whose coordinates satisfy
x1 x2 x23 3&3&3 x2n
0
is an ovoid.
4.5
Projective lines
A projective line over a field F has no non-trivial structure as an incidence
geometry. From the Kleinian point of view, though, it does have geometric structure, derived from the fact that the group PGL 2 F operates on it. As we saw
earlier, the action of this group is 3-transitive (sharply so if F is commutative),
and can even be 4-transitive for special skew fields of characteristic 2. However,
we assume in this section that the field is commutative.
It is conventional to label the points of the projective line over F with elements
of F ,]- ∞ / , as follows: the point 1 α is labelled by α, and the point 0 1 by ∞. (If we regard points of PG 2 F as lines in the affine plane AG 2 F , then
the label of a point is the slope of the corresponding line.)
Since PGL 2 F is sharply 3-transitive, distinguishing three points must give
unique descriptions to all the others. This is conveniently done by means of the
cross ratio, the function from 4-tuples of distinct points to F <^- 0 1 / , defined by
f x1 x2 x3 x4 C
x1 x3 x4 x2 x1 x4 x3 x2 3
60
4. Various topics
In calculating cross ratio, we use the same conventions for dealing with ∞ as
when elements of PGL 2 F are represented by linear fractional transformations;
for example, ∞ α ∞, and α∞ % β∞ α % β. Slightly differing forms of the cross
ratio are often used; the one given here has the property that f ∞ 0 1 α _ α.
Proposition 4.20 The group of permutations of PG 1 F preserving the cross ratio is PGL 2 F .
Proof Calculation establishes that linear fractional transformations do preserve
cross ratio. Also, the cross ratio as a function of its fourth argument, with the first
three fixed, is one-to-one, so a permutation which preserves cross ratio and fixes
three points is the identity. The result follows from these two assertions.
The cross ratio of four points is unaltered if the arguments are permuted in
two cycles of length 2: for example, f x3 x4 x1 x2 C f x1 x2 x3 x4 . These permutations, together with the identity, form a normal subgroup of index six in the
symmetric group S4 . Thus, in general, six different values are obtained by permuting the arguments. If α is one of these values, the others are 1 α, 1 % α, α 1 &% α,
1 % 1 α , and α % α 1 . There are two special cases where the number of values
is smaller, that is, where two of the six coincide. The relevant sets are - 1 2 12 / ,
and - ω & ω2 / , where ω is a primitive cube root of unity. A quadruple of points
is called harmonic if its cross ratios belong to the first set, equianharmonic if they
belong to the second. The first type occurs over any field of characteristic different from 2, while the second occurs only if F contains primitive cube roots of 1.
(But note that, if F has characteristic 3, then the two types effectively coincide:
1 2 12 , and the cross ratio of a harmonic quadruple is invariant under all
permutations of its arguments!)
In the arguments below, we regard a “quadruple” as being an equivalence class
of ordered quadruples (all having the same cross-ratio). So, for example, a harmonic quadruple (in characteristic different from 3) is a 4-set with a distinguished
partition into two 2-sets.
Proposition 4.21 Suppose that the characteristic of F is not equal to 2. Then the
group of permutations which preserve the set of harmonic quadruples is PΓL 2 F .
Proof Again, any element of PΓL 2 F preserves the set of harmonic quadruples. To see the converse, note that PGL 2 F contains a unique conjugacy class
4.5. Projective lines
61
of involutions having two fixed points, and that, if x1 x2 are fixed points and
x3 x4 a 2-cycle of such an involution, then - x1 x2 x3 x4 / is harmonic (and the
disthinguished partition is -H- x1 x2 / - x3 x4 /H/ ). Thus, these involutions can be reconstructed from the set of harmonic quadruples. So any permutation preserving
the harmonic quadruples normalises the group G generated by these involutions.
We see below that G is PSL 2 F if F contains square roots of 1, or contains this
group as a subgroup of index 2 otherwise. The normaliser of G is thus PΓL 2 F ,
as required.
(PSL n F is the group induced on the projective space by the invertible linear
transformations with determinant 1.)
We look further at the claim about G in the above proof. A transvection is a
linear transformation g with all eigenvalues equal to 1, for which ker g 1 has
codimension 1. In our present case, any 2 ` 2 upper unitriangular matrix different
from the identity is a transvection. The collineation of projective space induced
by a transvection is called an elation. An elation is characterised by the fact that
its fixed points form a hyperplane, known as the axis of the elation. Dually, an
elation fixes every line through a point, called the centre of the elation, which is
incident with the axis. In the present case n 2, the centre and axis of an elation
coincide.
Proposition 4.22 The elations in PGL 2 F generate PSL 2 F .
Proof The elations fixing a specified point, together with the identity, form a
group which acts sharply transitively on the remaining points. Hence the group
generated by the elations is 2-transitive. If α R 1 1 % β and γ α % β, then
a
1 1
0 1b
a
1 0
α 1b
a
1 β
0 1b
a
1 0
γ 1b
a
1% β
0
0
βb
so the two-point stabiliser in the group generated by all the elations contains that
in PSL 2 F . But elations have determinant 1, and so the group they generate is
a subgroup of PSL 2 F . So we have equality.
Now, if two distinct involutions have a common fixed point, then their product is a elation. Since all elations are conjugate, all can be realised in this way.
Thus the group G in the proof of Proposition 4.21 contains all elations, and hence
contains PSL 2 F .
62
4. Various topics
We conclude with a different way of giving structure to the projective line.
Suppose that E is a subfield of F. Then - ∞ / , E is a subset of the projective line
- ∞ / , F having the structure (in any of the senses previously defined) of projective line over E. We call any image of this set under an element of PGL 2 F a
circle. Then any three points lie in a unique circle. The points and circles form an
incidence structure which is an extension of the point-line structure of affine space
AG n E , where n is the degree of F over E. (For consider the blocks containing
∞. On removing the point ∞, we can regard F as an E-vector space of rank n; E
itself is an affine line, and the elements of PGL 2 F fixing ∞ are affine transformations; so, for any circle C containing ∞, C <\- ∞ / is an affine line. Since three
points lie in a unique circle, every affine line arises in this way.)
Sometimes, as we will see, this geometry can be represented as the points and
plane sections of a quadric over E. the most familiar example is the Riemann
sphere, which is the projective line over c , and can be identified with a sphere in
real 3-space so that the “circles” are plane sections.
4.6
Generation and simplicity
In this section, we extend to arbitrary rank the statement that PSL n F is
generated by elations, and show that this group is simple, except in two special
cases.
As before, F is a commutative field.
Theorem 4.23 For any n
W
2, the group PSL n F is generated by all elations.
Proof We use induction on n, the case n 2 having been settled by Proposition 4.22. The induction is based on the fact that, if W is a subspace of the axis of
an elation g, then g induces an elation on the quotient projective space modulo W .
Given g PSL n F , with g ! 1, we have to express g as a product of elations.
We may suppose that g fixes a point x. (For, if xg y ! x, and h is any elation
mapping x to y, then gh P 1 fixes x, and gh P 1 is a product of elations if and only if
g is.
By induction, we may multiply g by a product of elations (whose axes contain
x) to obtain an element fixing every line through x; so we may assume that g
itself does so. Considering a matrix representing g, and using the fact that g PSL n F , we see that g is an elation.
Theorem 4.24 Suppose that either n W 3, or n 2 and F 7 3. Then any nontrivial normal subgroup of PGL n F contains PSL n F .
4.6. Generation and simplicity
63
Proof We begin with an observation — if N is a normal subgroup of G, and
g N, g1 G, then N g g1 Od N, where N g g1 Oe g P 1 g1P 1 gg 1 is the commutator
of g and g1 — and a lemma:
Lemma 4.25 Under the hypotheses of Theorem 4.24, if g PGL n F maps the
point p1 of PG n 1 F to the point p2 , then there exists g1 PGL n F which
fixes p1 and p2 and doesn’t commute with g.
Proof Case 1: p2 g p3 ! p2 . We can choose g1 to fix p1 and p2 and move p3 .
(If p1 p2 p3 are not collinear, this is clear. If they are collinear, use the fact that
PGL 2 F is 3-transitive on the projective line, which has more than three points.
Case 2: p2 g p1 . Then g fixes the line p1 p2 , and we can choose coordinates
on this line so that p1 ∞, p2 0. Now g acts as x f> α % x for some α F. Let g1
induce x f> βx on this line; then N g g1 O induces x f> β2 x. So choose β ! 0 1 & 1,
as we may since F 3.
So let N be a non-trivial subgroup of PGL n F . Suppose that g N maps the
hyperplane H1 to H2 ! H1 . By the dual form of the Lemma, there exists g1 fixing
H1 and H2 and not commuting with g; then N g g1 O fixes H2 . So we may assume
that g N fixes a hyperplane H.
Next, suppose that g doesn’t fix H pointwise. The group of elations with
axis H is isomorphic to the additive group of a vector space whose associated
projective space is H; so there is a transvection g1 with axis H not commuting with
g. Then N g g1 O fixes H pointwise. So we may assume that g fixes H pointwise.
If g is not an elation, then it is a homology (induced by a diagonalisable linear
map with two eigenvalues, one having multiplicity n 1; equivalently, its fixed
points form a hyperplane and one additional point). Now if g1 is an elation with
axis H, then N g g1 O is a non-identity elation.
We conclude that N contains an elation. But then N contains all elations (since
they are conjugate), whence N contains PSL n F .
For small n and small finite fields F GF q , the group PSL n q g PSL n F is familiar in other guises. For n 2, recall that it is sharply 3-transitive of degree
q 1. Hence we have PSL 2 2 ih S3 , PSL 2 3 ih A4 , and PSL 2 4 Fh A5 (the
alternating groups of degrees 4 and 5 — the former is not simple, the latter is the
unique simple group of order 60). Less obviously, PSL 2 5 h A5 , since it is also
simple of order 60. Furthermore, PSL 2 7 jh PSL 3 2 (the unique simple group
of order 168), PSL 2 9 _h A6 , and PSL 4 2 _h A8 (for reasons we will see later).
64
4. Various topics
There has been a lot of work, much of it with a very geometric flavour, concerning groups generated by subsets of the set of elations. For example, McLaughlin [22, 23] found all irreducible groups generated by “full elation subgroups” (all
elations with given centre and axis). This result was put in a wider context by
Cameron and Hall . (In particular, they extended the result to spaces of infinite dimension.) Note that an important ingredient in the arguments of Cameron
and Hall is Theorem 4.9: under slight additional hypotheses, the set of all elation
centres satisfies the conditions on a colour class in that theorem. The result of
Theorem 4.9, together with the irreducibility of the group, then implies that every
point is an elation centre.
Exercises
1. (a) Prove that the non-negative integer m is the number of fixed points of
an element of PGL n q if and only if, when written in the base q, its digits are
non-decreasing and have sum not exceeding n.
(b) (Harder) Prove that the non-negative integer m is the number of fixed
points of an element of PΓL n F if and only if there exists r such that q is a
power of r and, when m is written in the base r, its digits are non-decreasing and
have sum at most n.
2. Prove that a simple group of order 60 possesses five Sylow 2-subgroups,
which it permutes by conjugation; deduce that such a group is isomorphic to A5 .
3. Modify the proof of Theorem 4.6.2 to show that, under the same hypotheses,
PSL n F is simple. [It is only necessary to show that the various g1 s can be
chosen to lie in PSL n F . The only case where this fails is Case 2 of the Lemma
when n 2, F GF 5 .]
4. (a) Let Π be a projective plane of order 4 containing a hyperoval X (six
points, no three collinear). Prove that there are natural bijections between the set
of lines meeting X in two points and the set of 2-subsets of X ; and between the
set of points outside X and the set of partitions of X into three 2-subsets. Find a
similar description of a set bijective with the set of lines disjoint from X . Hence
show that Π is unique (up to isomorphism).
(b) Let Π be a projective plane of order 4. Prove that any four points, no
three collinear, are contained in a hyperoval. Hence show that there is a unique
projective plane of order 4 (up to isomorphism).
(See Cameron and Van Lint [F] for more on the underlying combinatorial
principle.)
5
Buekenhout geometries
Francis Buekenhout introduced an approach to geometry which has the advantages of being both general, and local (a geometry is studied via its residues of
small rank). In this chapter, we introduce Buekenhout’s geometries, and illustrate with projective spaces and related objects. Further examples will occur later
(polar spaces).
5.1
Buekenhout geometries
So far, nothing has been said in general about what a “geometry” is. Projective
and affine geometries have been defined as collections of subspaces, but even the
structure carried by the set of subspaces was left a bit vague (except in Section 3.4,
where we used the inclusion partial order to characterise generalised projective
spaces as lattices). In this section, I will follow an approach due to Buekenhout
(inspired by the early work of Tits on buildings).
Before giving the formal definition, let us remark that the subspaces or flats
of a projective geometry are of various types (i.e., of various dimensions); may
or may not be incident (two subspaces are incident if one contains the other); and
are partially ordered by inclusion. To allow for duality, we do not want to take the
partial order as basic; and, as we will see, the betweenness relation derived from
it can be deduced from the type and incidence relations. So we regard type and
incidence as basic.
A geometry, or Buekenhout geometry, then, has the following ingredients: a
set X of varieties, a symmetric incidence relation I on X , a finite set ∆ of types,
∆. We require the following axiom:
and a type map τ : X
65
66
5. Buekenhout geometries
(B1) Two varieties of the same type are incident if and only if they are equal.
In other words, a geometry is a multipartite graph, where we have names for the
multipartite blocks (“types”) of the graph. We mostly use familiar geometric language for incidence; but sometimes, graph-theoretic terms like diameter and girth
will be useful. But one graph-theoretic concept is vital; a geometry is connected
if the graph of varieties and incidence is connected.
The rank of a geometry is the number of types.
A flag is a set of pairwise incident varieties. It follows from (B1) that the
members of a flag have different types. A geometry satisfies the transversality
condition if the following strengthening of (B1) holds:
(B2) (a) Every flag is contained in a maximal flag.
(b) Every maximal flag contains one variety of each type.
All geometries here will satisfy transversality.
Let F be a flag in a geometry G. The residue GF
follows: the set of varieties is
R F
of F is defined as
x X F : xIy for all y F ;
∆ τ F ; and incidence and the type map are the restricXF
the set of types is ∆F
tions of those in G. It satisfies (B1) (resp. (B2)) if G does. The type of a flag
or residue is its image under the type map, and the cotype is the complement of
the type in ∆; so the type of GF is the cotype of F. The rank and corank are the
cardinalities of the type and cotype.
A transversal geometry is called thick (resp. firm thin) if every flag of corank
1 is contained in at least three (resp. at least two, exactly two) maximal flags.
A property holds residually in a geometry if it holds in every residue of rank at
least 2. (Residues of rank 1 are sets without structure.) In particular, all geometries
of interest are residually connected; in effect, we assume residual connectedness
as an axiom:
(B3) All residues of rank at least 2 are connected.
The next result illustrates this concept.
Proposition 5.1 Let G be a residually connected transversal geometry, and let x
and y be varieties of X , and i and j distinct types. Then there is a path from x to y
in which all varieties except possibly x and y have type i or j.
5.1. Buekenhout geometries
67
Proof The proof is by induction on the rank. For rank 2, residual connectedness
is just connectedness, and the result holds by definition. So assume the result for
all geometries of smaller rank than G.
We show first that a two-step path whose middle vertex is not of type i or j
can be replaced by a path of the type required. So let xzy be a path of length 2.
Then x and y lie in the residue of z; so the assertion follows from the inductive
hypothesis.
Now this construction reduces by one the number of interior vertices not of
type i or j on a path with specified endpoints. Repeating it as often as necessary
gives the result.
The heart of Buekenhout’s idea is that “local” conditions on (or axiomatisations of) a geometry are really conditions about residues of small rank. This
motivates the following definition of a diagram.
Let ∆ be a finite set. Assume that, for any distinct i j ∆, a class i j of
geometries of rank 2 is given, whose two types of varieties are called “points”
and “blocks”. Suppose that the geometries in ji are the duals of those in i j .
The set ∆ equipped with these collections of geometries is called a diagram. It is
represented pictorially by taking a “node” for each element of ∆, with an “edge”
between each pair of nodes, the edge from i to j being adorned or labelled with
some symbol for the class i j . We will see examples later.
A geometry G belongs to the diagram ∆ i j : i j ∆ if ∆ is the set of types
of G and, for all distinct i j ∆, and all residues GF in G with rank 2 and type
i j , GF is isomorphic to a member of i j (where we take points and blocks in
GF to be varieties of types i and j respectively).
In order to illustrate this idea, we need to define some classes of rank 2 geometries to use in diagrams. Some of these we have met already; but the most
important is the most trivial: A digon is a rank 2 geometry (having at least two
points and at least two blocks) in which any point and block are incident; in other
words, a complete bipartite graph containing a cycle. By abuse of notation, the
“labelled edge” used to represent digons is the absence of an edge! This is done
in part because most of the rank 2 residues of our geometries will be digons, and
this convention leads to uncluttered pictorial representations of diagrams.
A partial linear space is a rank 2 geometry in which two points lie on at most
one line (and dually, two lines meet in at most one point). It is represented by an
edge with the label Π, thus:
Π
68
5. Buekenhout geometries
We already met the concepts linear space and generalised projective plane: they
are partial linear spaces in which the first, resp. both, occurrences of “at most”
are replaced by “exactly”. They are represented by edges with label L and without any label, respectively. (Conveniently, the labels for the self-dual concepts
of “partial linear space” and “generalised projective plane” coincide with their
mirror-images, while the label for “linear space” does not.) Note that a projective plane is a thick generalised projective plane. Another specialisation of linear
spaces, a “circle” or “complete graph”, has all lines of cardinality 2; it is denoted
by an edge with label c.
Now we can give an example:
Proposition 5.2 A projective geometry of dimension n has the diagram
Proof Transversality and residual connectivity are straightforward to check. We
verify the rank 2 residues. Take the types to be the dimensions 0 1
n 1, and
let F be a flag of cotype i j , where i j.
Case 1:
j
i
1. Then F has the form
U U U Its residue consists of all subspaces of dimension i or i 1 between U U0
U1
i 1
i 2
n 1
and Ui 2 ;
2 Ui 1 .
i 1
this is clearly the projective plane based on the rank 3 vector space Ui
i 1. Now the flag F looks like
U U U U U U Its residue consists of all subspaces lying either between U and U , or between U and U . Any subspace X of the first type is incident with any subspace Y of the second, since X U U Y . So the residue is a digon.
Case 2:
j
0
i 1
i 1
j 1
j 1
n 1
i 1
j 1
i 1
j 1
i 1
j 1
In diagrams, it is convenient to label the nodes with the corresponding elements of ∆. For example, in the case of a projective geometry of dimension n, we
take the labels to be the dimensions of varieties represented by the nodes, thus:
0
1
2
n 2
n 1
5.1. Buekenhout geometries
69
I will use the convention that labels are placed above the nodes where possible.
This reserves the space below the nodes for another use, as follows.
A transversal geometry is said to have orders, or parameters, if there are numbers si (for i ∆) with the property that any flag of cotype i is contained in exactly
si 1 maximal flags. If so, these numbers si are the orders (or parameters). Now,
if G is a geometry with orders, then G is thick/firm/thin respectively if and only
if all orders are 1/ 1/ 1 respectively. We will write the orders beneath the
nodes, where appropriate. Note that a projective plane of order n (as defined earlier) has orders n n (in the present terminology). Thus, the geometry PG n q has
diagram
0
q
1
q
2
n 2
n 1
q
q
q
We conclude this section with some general results about Buekenhout geometries. These results depend on our convention that a non-edge symbolises a digon.
Proposition 5.3 Let the diagram ∆ be the disjoint union of ∆1 and ∆2 , with no
edges between these sets. Then a variety with type in ∆1 and one with type in ∆2
are incident.
Proof We use induction on the rank. For rank 2, ∆ is the diagram of a digon,
2, and (without loss of
and the result is true by definition. So assume that ∆
generality) that ∆1
1.
Let Xi be the set of varieties with type in ∆i , for i 1 2. By the inductive
hypothesis, if x y X1 with xIy, then R x X2 R y X2 . (Considering R x ,
we see that every variety in R x X2 is incident with y, so the left-hand set is
contained in the right-hand set. Reversing the rôles of x and y establishes the
result.) Now by connectedness, R x X2 is independent of x X1 . (Note that
Proposition 5.1 is being used here.) But this set must be X2 , since every variety in
X2 is incident with some variety in X1 .
#!
"!
"!
\$!
A diagram is linear if the “non-digon” edges form a simple path, as in the
diagram for projective spaces in Proposition 5.3 above.
Suppose that one particular type in a geometry is selected, and varieties of that
type are called points. Then the shadow, or point-shadow, of a variety x is the set
Sh x of varieties incident with x. Sometimes we write Sh0 x , where 0 is the type
of a point. In a geometry with a linear diagram, the convention is that points are
varieties of the left-most type.
70
5. Buekenhout geometries
&%
Corollary 5.4 In a linear diagram, if xIy, and the type of y is further to the right
Sh y .
than that of x, then Sh x
Proof R x has disconnected diagram, with points and the type of y in different
components; so, by Proposition 5.2, every point in R x is incident with y.
Exercises
1. (a) Construct a geometry which is connected but not residually connected.
(b) Show that, if G has any of the following properties, then so does any residue
of G of rank at least 2: residually connected, transversal, thick, firm, thin.
2. Show that any generalised projective geometry belongs to the diagram
'
'
3. (a) A chamber of a transversal geometry G is a maximal flag. Let be the
set of chambers of the geometry G. Form a graph with vertex set
by joining
two chambers which coincide in all but one variety. G is said to be chamberconnected if this graph is connected. Prove that a residually connected geometry
is chamber-connected, and a chamber-connected geometry is connected.
(b) Consider the 3-dimensional affine space AG 3 F over the field F. Take
three types of varieties: points (type 0), lines (type 1), and parallel classes of
planes (type 2). Incidence between points and lines is as usual; a line L and
a parallel class C of planes are incident if L lies in some plane of C; and any
variety of type 0 is incident with any variety of type 2. Show that this geometry is
chamber-connected but not residually connected.
(c) Let V be a six-dimensional vector space over a field F, with a basis e1 e2 e3 f1 f2 f3 .
Let G be the additive group of V , and let H1 , H2, H3 be the additive groups of the
three subspaces e2 e3 f1 , e3 e1 f2 , and e1 e2 f3 . Form the coset geometry G H1 H2 H3 : its vaarieties of type i are the cosets of Hi in G, and two
varieties are incident if and only if the corresponding cosets have non-empty intersection. Show that this geometry is connected but not chamber-connected.
( ) ( )
5.2
( )
Some special diagrams
In this section, we first consider geometries with linear diagram in which all
strokes are linear spaces; then we specialise some or all of these linear spaces to
projective or affine planes. We will see that the axiomatisations of projective and
affine spaces can be expressed very simply in this formalism.
5.2. Some special diagrams
Theorem 5.5 Let G be a geometry with diagram
0 L 1 L 2
n 2 L
71
n 1
Let varieties of type 0 and 1 be points and lines.
*
(a) The points and shadows of lines form a linear space .
&%
*
(b) The shadow of any variety is a subspace of .
(c) Sh0 x
Sh0 y if and only if x is incident with y.
(d) If x is a variety and p a point not incident with x, then there is a unique variety
τ x 1.
y incident with x and p such that τ y
Proof (a) We show that two points lie on at least one line by induction on the
rank. There is a path between any two points using only points and lines, by
Proposition 5.2; so it suffices to show that any such path of length greater than 2
can be shortened. So assume pILIqIMIr, where p q r are points and L M lines.
By the induction hypothesis, the POINTs L and M of R q lie in a LINE Π, a
plane of G incident with L and M. By Corollary 5.4, p and q are incident with Π.
Since Π is a linear space, there is a line through p and q. (The convention of using
capitals for varieties in R q is used here.)
Now suppose that two lines L and M contain the two points p and q. Considering R p , we find a plane Π incident with L and M and hence with p and q. But
Π is a linear space, so L M.
(b) Let y be any variety, and p q Sh0 y . Since points and lines incident
with y form a linear space by (a), there is a line incident with p q and y. This must
be the unique line incident with p and q; and, by Corollary 5.4, all its points are
incident with y and so are in Sh0 y .
(c) The reverse implication is Corollary 5.4. So suppose that Sh0 x
Sh0 y .
Take p Sh0 x . Then, in R p , we have Sh1 x
Sh1 y (since these shadows
are linear subspaces) , and so xIy by induction. (The base case of the induction,
where x is a line, is covered by (b).)
(d) This is clear if x is a point. Otherwise, choose q Sh0 x , and apply
induction in R q (replacing p by the line pq).
+%
%
Theorem 5.6 A geometry with diagram
,
is a generalised projective space (of finite dimension).
72
5. Buekenhout geometries
Proof By Theorem 5.5(d), a potential Veblen configuration lies in a plane; since
planes are projective, Veblen’s axiom holds. It remains to show that every linear
subspace is the shadow of some variety; this follows easily by induction.
Theorem 5.7 A geometry with diagram
L
consists of the points, lines and planes of a (possibly infinite-dimensional) generalised projective space.
Proof Veblen’s axiom is verified as in Theorem 5.6. It is clear that every point,
line or plane corresponds to a variety.
Remark. Consider geometries with the diagram
L
-
L
By the argument for Theorem 5.7, we have all the points, lines and planes, and
some higher-dimensional varieties, of a generalised projective space. Examples
arise by taking all the flats of dimension at most r 1, where r is the rank. However, there are other examples. A simple case, with r 4, can be constructed as
follows.
Let
be a projective space of countable dimension over a finite field F.
Enumerate the 3-dimensional and 4-dimensional subspaces in lists T0 T1
and
F0 F1
. Now construct a set
of 4-dimensional subspaces in stages as follows. At the nth stage, if Tn is already contained in a member of , do nothing.
Otherwise, of the infinitely many subspaces Fj which contain Tn , only finitely
many are excluded because they contain any Tm with m n; let Fi be the one with
smallest index which is not excluded, and adjoin it to . At the conclusion, any
3-dimensional subspace is contained in a unique member of . Then the points,
lines, planes, and subspaces in form a geometry with the diagram
.
'
'
'
L
L
'
'
where the first L denotes the points and lines in 3-dimensional projective space
over F.
5.2. Some special diagrams
73
Now we turn to affine spaces, where similar results hold. The label A f on a
stroke will denote the class of affine planes.
Theorem 5.8 A geometry with diagram
Af
,
is an affine space of finite dimension.
Proof It is a linear space whose planes are affine (that is, satisfying condition
(AS1) of Section 11). We must show that parallelism is transitive. So suppose
that L1 L2 L3 , but L1 L3 . Then all three lines lie in a subspace of dimension
3; so it is enough to deduce a contradiction in the case of geometries of rank 3.
, two planes which have a
Note that, for a geometry with diagram A f
common point must meet in a line.
Let Π1 be the plane through L1 and L2 , and Π2 the plane through p and L3 ,
where p is a point of L1 . Then Π1 and Π2 both contain p, so they meet in a line
M L1 . Then M is not parallel to L2 , so meets it in a point q, But then Π2 contains
L3 and q, hence L2 , and so is equal to Π1 , a contradiction.
The fact that all linear subspaces are shadows of varieties is proved as in Theorem 5.6.
/ /
01/
0
Theorem 5.9 A geometry with diagram
Af
L
in which some line has more than three points, consists of the points, lines and
planes of a (possibly infinite-dimensional) affine space.
The proof is as for Theorem 5.7, using Buekenhout’s Theorem 3.10.
Exercises
1. Consider a geometry of rank n with diagram
L
,
in which all lines have the same finite cardinality k, and all the projective planes
have the same finite order q.
74
5. Buekenhout geometries
(a) If n 4, prove that the geometry is either projective (q
(q k).
(b) If n 3, prove that q k 1 k k2 or k k2 1 .
(This result is due to Doyen and Hubaut ).
2. Construct an infinite “free-like” geometry with diagram
c
k
1) or affine
(Ensure that three points lie in a unique plane, while two planes meet in two
points.)
c
Af .
3. (a) Show that an inversive plane belongs to the diagram
What are the varieties?
(b) Show how to construct a geometry with diagram
c
Af
(n nodes) from an ovoid in PG n F (see Section 4.4).
6
Polar spaces
Now we begin on our second major theme, polar spaces. This chapter corresponds
to the first half of Chapter 1, and gives the algebraic description of polar spaces.
The algebraic background required is more elaborate (vector spaces with forms,
rather than just vector spaces), accounting for the increased length. The first section, on polarities of projective spaces, provides motivation for the introduction of
the (Hermitian and quadratic) forms.
6.1
Dualities and polarities
Recall that the dual V of a finite-dimensional (left) vector space V over a
skew field F can be regarded as a left vector space of the same dimension over
the opposite field F , and
there is thus an inclusion-reversing bijection between
the projective spaces PG n F and PG n F . If it happens that F and F are isomorphic,
then there exists a duality of PG n F , an inclusion-reversing bijection
of PG n F .
Conversely, if PG n F admits a duality (for n 1), then F is isomorphic to
F , as follows from the FTPG (see Section 1.3). We will examine this conclusion
and make it more detailed.
So let π be a duality of PG
n F , n 1. Composing π with the natural isomorphism from
PG n F to PG
n F , we obtain an inclusion-preserving bijection θ
from PG n F to PG n F . According to the FTPG, θ is induced by a semilinear
transformation T from V F n 1 to its dual space V , associated with an isomor
phism σ : F F , which can be regarded as being an anti-automorphism of F:
75
76
6. Polar spaces
that is,
v1 v2 T
αv T
v1 T v2 T ασ vT V F by the rule
b v w v wT Define a function b : V
that is, the result of applying the element wT of V to v. Then b is a sesquilinear
form: it is linear as a function of the first argument, and semilinear as a function
of the second — this means that
b v w2 b v w1 w2 b v w1 and
b v αw ασ b v w (The prefix “sesqui-” means “one-and-a-half”.) If we need to emphasise the antiautomorphism σ, we say that b is σ-sesquilinear. If σ is the identity, then the form
is bilinear.
The form b is also non-degenerate, in the sense that
and
w V b v w 0
v
V b v w 0
v 0
w 0
(The second condition asserts that T is one-to-one, so that if w 0 then wT is
a non-zero functional. The first asserts that T is onto: only the zero vector is
annihilated by every functional in the dual space.)
So, we have:
Theorem 6.1 Any duality of PG n F , for n 1, is induced by a non-degenerate
σ-sesquilinear form on the underlying vector space, where σ is an anti-automorphism
of F.
Conversely, any non-degenerate sesquilinear form on V induces a duality. We
can short-circuit the passage to the dual space, and write the duality as
U
U v V : b v w 0 for all w U 6.1. Dualities and polarities
77
Obviously, a duality applied twice is a collineation. The most important
types
of dualities are those whose square is the identity. A polarity of PG n F is a
duality which satisfies U U for all flats U of PG n F .
It is a bit difficult to motivate the detailed study of polarities at this stage; but
it will turn out that they give rise to a class of geometries (the polar spaces) with
properties similar to those of projective spaces. To put it somewhat vaguely, we
are trying to add some extra structure to a projective space; if a duality is not
a polarity, then its square is a non-identity collineation, and some of the extra
structure arises from this collineation. Only in the case of a polarity is the extra
structure “primitive”.
A sesquilinear form b is reflexive if b v w 0 implies b w v 0.
Proposition 6.2 A duality is a polarity if and only if the sesquilinear form defining
it is reflexive.
Proof b is reflexive if and only if
v w !"
w v !#
Hence, if b is reflexive, then U \$ U for all subspaces U . But by non-degeneracy,
dimU % dimV & dimU dimU ; and so U U % for all U . Conversely,
given a polarity , if w ' v , then v ( v % \$) w (since inclusions are
reversed).
We now turn to the classification of reflexive forms. For convenience, from
now on F will always be assumed to be commutative. (Note that, if the antiautomorphism σ is an automorphism, and in particular if σ is the identity, then F
is automatically commutative.)
The form b is said to be σ-Hermitian
if b w v * b v w σ for all v w V .
This implies that, for any v, b v v lies in the fixed field of σ. If σ is the identity,
such a form (which is bilinear) is called symmetric.
A bilinear form b is called alternating if b v v % 0 for all v V . This implies
that b w v +,& b v w for all v w V . (Expand b v w v w - 0, and note
that two of the four terms are zero.) Hence, if the characteristic is 2, then any
alternating form is symmetric (but not conversely); but, in characteristic different
from 2, only the zero form is both symmetric and alternating.
Clearly, an alternating or Hermitian form is reflexive. Conversely, we have the
following:
78
6. Polar spaces
Theorem 6.3 A non-degenerate reflexive σ-sesquilinear form is either alternating, or a scalar multiple of a σ-Hermitian form. In the latter case, if σ is the
identity, then the scalar can be taken to be 1.
I will not give the complete proof of this theorem. The next result shows that
1, and then the proof of the theorem is given in the case of a bilinear form
(that is, when σ 1).
σ2
Proposition 6.4 If b is a non-zero reflexive σ-sesquilinear form, then σ2 is the
identity.
Proof Note first that a form is σ-sesquilinear if and only if it is additive in each
variable and satisfies
b v βw b v w βσ b αv w αb v w .
Step
1 If b is alternating, then σ 1. For we can choose v and w with b v w /
& b w v 1. Then for any α F, we have
α
αb v w b αv w & b w αv & b w v ασ
ασ (Note that this step does not require non-degeneracy, merely that b is not identically zero.)
So we can assume that there exists v with b v v 0 0. Multiplying b by a nonzero scalar (this does not affect the hypotheses), we may assume that b v v 1.
Step 2 Assume for a contradiction that σ2 1. For any vector w, if b w v 1 0,
then we can replace w by its product
with a non-zero scalar
to assume b w v / 1.
Then
b w & v v 0, and so b v w & v 0, whence b v w 1. We claim that
b w w 1.
6.1. Dualities and polarities
79
Proof
Suppose that α b w w 2 1. Note first that b w & αv v - 0, and
so b w w & αv + 0, whence α ασ . Take any element λ F with λ 1, and
choose µ F such that µσ 1 & λ 43 1 α & λ . Since α 1, we have µ 1; and
This implies, first, that λ µσ & λµσ
α & λ
α & µσ 1 & µσ 43 1 , and second that
b w & λv w & µv α & λ & µσ λµσ
0
Hence b w & µv w & λv 0, and we obtain
α & µ & λσ µλσ
0
Applying σ to this equation and using the fact that ασ
α & µσ & λ σ
λ σ µσ 0 2
whence
λσ
2
α, we obtain
2
α & µσ 1 & µσ 3 1 λ
But λ was an arbitrary element different from 1. Since clearly 1σ
σ2 1, contrary to assumption.
1, we have
Step 3 Let W v . Then V 5 v 76 W , and rk W 98 1. For
any x W , we
have b v v 0 b v x v : 1, and so by Step 2, we have b v x v x : 1.
Thus b x x +;& 2. Putting x 0, we see that F must have characteristic 2, and
that b < W is alternating. But then Step 1 shows that b < W is identically zero, whence
W is contained in the radical of b, contrary to the assumed non-degeneracy.
Proof of Theorem 6.3 We have
b u v b u w /& b u w b u v % 0
by commutativity; that is, using bilinearity,
b u b u v w & b u w v 0 By reflexivity,
b b u v w & b u w v u 0 80
6. Polar spaces
whence bilinearity again gives
b u v b w u b u w b v u (6.1)
Call
a vector u good if b u v b v u 1 0 for some v. By (6.1), if u is good,
then b u w % b w u for all w. Also, if u is good and b u v : 0, then
v is good.
But, given any two non-zero vectors u1 u2 , there exists v with b ui v = 0 for
i 1 2. (For there exist v1 v2 with b ui vi 2 0 for i 1 2, by non-degeneracy;
and at least one of v1 v2 v1 v2 has the required property.) So, if some vector is
good, then every non-zero vector is good, and b is symmetric.
But, putting u w in (6.1) gives
b u u b u v /& b v u ! 0
for all u v. So, if u is not good, then b u u 0; and, if no vector is good, then b
is alternating.
In the next few sections, we develop this theme further.
Exercises
1. Let b be a sesquilinear form on V . Define the left and right radicals of b to
be the subsets
>
v V : w V b v w 0 and
>
v V : w V b w v 0 respectively. Prove that the left and right radicals are subspaces of the same rank
(if V has finite rank).
(Note: If the left and right radicals are equal, this subspace is called the radical
of b. This holds if b is reflexive.)
2. Give an example of a bilinear form on an infinite-rank vector space whose
left radical is zero and whose right radical is non-zero.
3. Let σ be a (non-identity) automorphism of F of order 2. Let E be the
subfield Fix σ .
(a) Prove that F is of degree 2 over E, i.e., a rank 2 E-vector space.
[See any textbook on Galois theory. Alternately,
argue as follows: Take λ F ? E. Then λ is quadratic over E, so E λ has degree 2 over E. Now E λ contains an element ω such that ωσ ;& ω (if the characteristic is not 2) or ωσ 6.2. Hermitian and quadratic forms
81
ω 1 (if the characteristic is 2). Now, given two such elements, their quotient or
difference respectively is fixed by σ, so lies in E.]
(b) Prove that
λ F : λλσ 1 @ ε A εσ : ε F [The left-hand set clearly contains the right. For the reverse inclusion, separate
into cases according as the characteristic is 2 or not.
If the characteristic is not 2, then we can take F E ω , where ω2 α E
and ω σ B& ω. If λ 1, then take ε 1; otherwise, if λ a bω, take ε bα a & 1 ω.
If the characteristic is 2, show that we can take F E ω , where ω2 ω α 0, α E, and ωσ ω 1. Again, if λ 1, set ε 1; else, if λ a bω, take
ε a 1 bω.]
4. Use
the result of Exercise 3 to complete the proof of Theorem 6.3 in general.
[If b u u C 0 for all u,
theσform b is alternating and bilinear. If not, suppose
that b u u D 0 and let b u u λb u u . Choosing ε as in Exercise 2 and renormalising b, show that we may assume that λ 1, and (with this choice) that b
is Hermitian.]
6.2
Hermitian and quadratic forms
We now change ground slightly from the last section. On the one hand, we
restrict things by excluding some bilinear forms from the discussion; on the other,
we introduce quadratic forms. The loss and gain exactly balance if the characteristic is not 2; but, in characteristic 2, we make a net gain.
of the commutative field F, of order dividing 2. Let
Let σ be an automorphism
σ
Fix σ +E λ F : λ λ be the fixed field of σ, and Tr σ CE λ λσ : λ F the trace of σ. Since σ2 is the
identity, it is clear that Fix σ *F Tr σ . Moreover,
if σ is the identity, then Fix σ F, and
Tr σ HG
0
F
if F has characteristic 2,
otherwise.
that b v v 1
Let b be a σ-Hermitian form. We observed in the last section
Fix σ for all v V . We call the form b trace-valued if b v v 1 Tr σ for all
v V.
Proposition 6.5 We have Tr σ + Fix σ unless the characteristic of F is 2 and
σ is the identity.
82
6. Polar spaces
Proof E Fix σ is a field, and K Tr σ is an E-vector space contained in E
(Exercise 1). So, if K E, then K 0, and σ is the map x & x. But, since σ is
a field automorphism, this implies that the characteristic is 2 and σ is the identity.
Thus, in characteristic 2, symmetric bilinear forms which are not alternating are not trace-valued; but this is the only obstruction. We introduce quadratic
forms to repair this damage. But, of course, quadratic forms can be defined in any
characteristic. However, we note at this point that Proposition 6.5 depends in a
crucial way on the commutativity of F; this leaves open the possibility of additional types of polar spaces defined by so-called pseudoquadratic forms. These
will be discussed briefly in Section 7.6.
Let V be a vector space over F. A quadratic form on V is a function f : V F
satisfying
I
I f λv λ2 f v for all λ F, v V ;
f v w f v f w b v w , where b is bilinear.
Now, if the characteristic of F is not 2, then b is a symmetric bilinear form.
Each of f and b determines the other, by
b v w and
f v w /& f v /& f w f v 1
2b
v v .
the latter equation coming from the substitution v w in the second defining
condition. So nothing new is obtained.
On the other hand, if the characteristic of F is 2, then b is an alternating bilinear
form, and f cannot be recovered from b. Indeed, many different quadratic forms
correspond to the same bilinear form. (Note that the quadratic form does give
extra structure to the vector space; we’ll see that this structure is geometrically
similar to that provided by an alternating or Hermitian form.)
We say that the bilinear form is obtained by polarisation of f .
Now let b be a symmetric
bilinear
form over a field of characteristic 2, which
is not alternating. Set f v b v v . Then we have
f λv λ2 f v and
f v w f v
f w .
6.2. Hermitian and quadratic forms
83
since b v w b w v J 0. Thus f is “almost” a semilinear form; the map λ λ2
is a homomorphism of the field F with kernel 0, but it may fail to be an automorphism. But in any case, the kernel of f is a subspace of V , and the restriction of
b to this subspace is an alternating bilinear form. So again, in the spirit of the
vague comment motivating the study of polarities in the last section, the structure
provided by the form b is not “primitive”. For this reason, we do not consider
symmetric bilinear forms in characteristic 2 at all. However, as indicated above,
we will consider quadratic forms in characteristic 2.
Now, in characteristic different from 2, we can take either quadratic forms or
symmetric bilinear forms, since the structural content is the same. For consistency,
we will take quadratic forms in this case too. This leaves us with three “types” of
forms to study: alternating bilinear forms; σ-Hermitian forms where σ is not the
identity; and quadratic forms.
We have to define the analogue of non-degeneracy for quadratic forms. Of
course, we could require that the bilinear form obtained by polarisation is nondegenerate; but this is too restrictive. We say that a quadratic form f is nonsingular if
f v 0 &
w V b v w 0 v 0
where b is the associated bilinear form; that is, if the form f is non-zero on every
non-zero vector of the radical.
If the characteristic is not 2, then non-singularity is equivalent to non-degeneracy
of the bilinear form.
Now suppose that the characteristic is 2, and let W be the radical. Then b is
identically zero on W ; so the restriction of f to W satisfies
f v w K
f λv L
f v f w λ2 f v .
As above, f is very nearly semilinear. The field F is called perfect if every element
is a square. In this case, f is indeed semilinear, and its kernel is a hyperplane of
W . We conclude:
Theorem 6.6 Let f be a non-singular quadratic form, which polarises to b, over
a field F.
(a) If the characteristic of F is not 2, then b is non-degenerate.
(b) If F is a perfect field of characteristic 2, then the radical of b has rank at
most 1.
84
6. Polar spaces
Exercises
1. Let σ be an automorphism of a commutative field F such that σ2 is the
identity.
(a) Prove that Fix σ is a subfield of F.
(b) Prove that
Tr σ is closed under addition, and under multiplication by
elements of Fix σ .
2. Let b be an alternating
bilinear form on a vector space V over a field F of
characteristic 2. Let vi : i I be a basis for V , and q any function from
I to F.
Show that there is a unique quadratic form with the properties that f vi # q i for every i I, and f polarises to b.
3. (a) Construct an imperfect field of characteristic 2.
(b) Construct a non-singular quadratic form with the property that the radical
of the associated bilinear form has rank greater than 1.
4. Show that finite fields of characteristic 2 are perfect. (Hint: the multiplicative group is cyclic of odd order.)
6.3
Classification of forms
As explained in the last section, we now consider a vector space V of finite
rank equipped with a form of one of the following types: a non-degenerate alternating bilinear form b; a non-degenerate σ-Hermitian form b, where σ is not the
identity; or a non-singular quadratic form f . In the third case, we let b be the bilinear form obtained by polarising f ; then b is alternating or symmetric according
as the characteristic is or is not 2, but b may be degenerate.
In the other two cases,
we define a function f : V F defined by f v 0 b v v — this is identically
zero if b is alternating. See Exercise 1 for the Hermitian case.
We say that V is anisotropic if f v M 0 for all v 0.
Also, V is a hyperbolic
line if it is spanned by vectors v and w with f v # f w # 0 and b v w - 1.
(The vectors v and w are linearly independent, so V has rank 2; so, projectively, it
is a “line”.)
Theorem 6.7 A space carrying a form of one of the above types is the direct sum
of a number r of hyperbolic lines and an anisotropic space U . The number r and
the isomorphism type of U are invariants of V .
Proof If V is anisotropic, then there is nothing to prove. (V cannot
contain a
hyperbolic line.) So suppose that V contains a vector v 0 with f v 0.
6.3. Classification of forms
85
We claim that there is a vector w with b v w N 0. In the alternating and
Hermitian cases, this follows immediately from the non-degeneracy of the form.
In the quadratic case, if no such vector exists, then v is in the radical of b; but v is
a singular vector, contradicting the non-singularity of f .
Multiplying w by a non-zero constant, we may assume that b v w 1.
v w & λv 1. We wish to choose λ so
Now,
for
any
value
of
λ,
we
have
b
that f w & λv 0; then v and w will span a hyperbolic line. Now we distinguish
cases. If b is alternating, then any value of λ works. If b is Hermitian, we have
f w & λv L
f w O& λb v w O& λσb w v f w O& λ λσ ;
and, since b is trace-valued, there exists λ with Tr λ 0
quadratic, we have
f w & λv L
f w O& λb w v f w O& λ λλσ f v f w . Finally, if f is
λ2 f v so we choose λ f w .
Now let W1 be the hyperbolic line v w & λv , and let V1 W1 , where orthogonality is defined with respect to the form b. It is easily checked that V V1 6 W1 ,
and the restriction of the form to V1 is still non-degenerate or non-singular, as
appropriate. Now the existence of the decomposition follows by induction.
I will omit the proof of uniqueness.
The number r of hyperbolic lines is called the polar rank or Witt index of V . I
do not know of a commonly accepted term for U ; I will call it the germ of V , for
reasons which will become clear shortly.
To complete the classification of forms over a given field, it is necessary to
determine all the anisotropic spaces. In general, this is not possible; for example, the study of positive definite quadratic forms over the rational numbers leads
quickly into deep number-theoretic waters. I will consider the cases of the real
and complex numbers and finite fields.
First, though, the alternating case is trivial:
Proposition 6.8 The only anisotropic space carrying an alternating bilinear form
is the zero space.
86
6. Polar spaces
In combination with Theorem 6.7, this shows that a space carrying a nondegenerate alternating bilinear form is a direct sum of hyperbolic lines.
Over the real numbers, Sylvester’s theorem asserts that any quadratic form in
n variables is equivalent to the form
x21 P
P
x2r & x2r 1 &
P
P
4& x2r s for some r s with r s Q n. If the form is non-singular, then r s n. If both r
and s are non-zero, there is a non-zero singular vector (with 1 in positions 1 and
r 1, 0 elsewhere). So we have:
Proposition 6.9 If V is a real vector space of rank n, then an anisotropic form
on V is either positive definite or negative definite; there is a unique form of each
type up to invertible linear transformation, one the negative of the other.
The reals have no non-identity automorphisms, so Hermitian forms do not
arise.
Over the complex numbers, the following facts are easily shown:
(a) There is a unique non-singular quadratic form (up to equivalence) in n
variables for any n. A space carrying such a form is anisotropic if and only if
n Q 1.
(b) If σ denotes complex conjugation, the situation for σ-Hermitian forms is
the same as for quadratic forms over the reals: anisotropic forms are positive or
negative definite, and there is a unique form of each type, one the negative of the
other.
For finite fields, the position is as follows.
Theorem 6.10 (a) An anisotropic quadratic form in n variables over GF q exists
if and only if n Q 2. There is a unique form for each n except when n 1 and q is
odd, in which case there are two forms, one a non-square multiple of the other.
(b) Let q r2 and let σ be the field automorphism α αr . Then there is
an anisotropic σ-Hermitian form in n variables if and only if n Q 1. The form is
unique in each case.
Proof (a) Consider
first the case where the characteristic is not 2. The multiplicative group of GF q is cyclic of even order q & 1; so the squares form a subgroup
of index 2, and if η is a fixed non-square, then every non-square has the form ηα2
for some α. It follows easily that any quadratic form in one variable is equivalent
to either x2 or ηx2 .
6.3. Classification of forms
87
Next, consider non-singular forms in two variables. By completing the square,
2 2
2
2
2
2
such a form is equivalent to one
of x y , x ηy , ηx ηy .
Suppose first that q R 1 mod 4 . Then & 1 is a square, say & 1 β2 . (In
the
multiplicative group, & 1 has order 2, so lies
in the subgroup of even order
1
2 y2 q
&
1
consisting
of
squares.)
Thus
x
x βy x & βy , and the first and
2
third forms are not anisotropic. Moreover, any form in 3 or more variables, when
converted to diagonal form, contains one of these two, and so is not anisotropic
either.
Now consider the other case, q RB& 1 mod 4 . Then & 1 is a non-square
(since the group of squares has odd order), so the second form is x y x & y ,
and is not anisotropic. Moreover, the set of squares is not
closed under addition
1
(else it would be a subgroup of the additive group, but 2 q 1 doesn’t divide q);
so there exist two squares whose sum is a non-square. Multiplying by a suitable
square, there exist β γ with β2 γ2 S& 1. Then
& x2 y2 + βx γy 2 γx & βy 2 and the first and third forms are equivalent. Moreover, a form in three variables
is certainly not anisotropic
unless it is equivalent to x2 y2 z2 , and this form
vanishes at the vector β γ 1 ; hence there is no anisotropic form in three or more
variables.
The characteristic 2 case is an exercise (see Exercise 3).
(b) Now consider Hermitian forms. If σ is an automorphism of GF q of order
2, then q is a square,
say q r2 , and ασ αr . We need the fact that every element
of Fix σ GF r has the form αασ (see Exercise 1 of Section 6.2). In one variable, we have f x - µxxσ for some non-zero µ Fix σ ; writing
µ αασ and replacing x by αx, we can assume that µ 1.
σ
σ
In two variables,σwe can similarly take the form
to be xx yy . Now & 1 Fix σ , so & 1 λλ ; then the form vanishes at 1 λ . It follows that there is no
anisotropic form in any larger number of variables either.
Exercises
1. Let b be a σ-Hermitian
form on a vector
space V over F, where σ is not the
identity. Set f v * b v v . Let E Fix σ , and let V T be V regarded as an Evector space by restricting scalars.
form
on V T , which
Prove that f is a quadratic
σ
polarises to the bilinear
form Tr b defined by Tr b v w b v w b v w .
Show further that Tr b is non-degenerate if and only if b is.
88
6. Polar spaces
2. Prove that there is, up to equivalence, a unique non-degenerate alternating
bilinear form on a vector space of countably infinite dimension (a direct sum of
countably many isotropic lines).
3. Let F be a finite field of characteristic 2.
(a) Prove that every element of F has a unique square root.
(b) By considering the bilinear form obtained by polarisation, prove that a
non-singular form in 2 or 3 variables over F is equivalent to αx2 xy βy2 or
αx2 xy βy2 γz2 respectively. Prove that forms of the first shape (with α β 0)
are all equivalent, while those of the second shape cannot be anisotropic.
6.4
Classical polar spaces
Polar spaces describe the geometry of vector spaces carrying a reflexive sesquilinear form or a quadratic form in much the same way as projective spaces describe
the geometry of vector spaces. We now embark on the study of these geometries;
the three preceding sections contain the prerequisite algebra.
First, some terminology. The polar spaces associated with the three types of
forms (alternating bilinear, Hermitian, and quadratic) are referred to by the same
names as the groups associated with them: symplectic, unitary, and orthogonal
respectively. Of what do these spaces consist?
Let V be a vector space carrying a form of one of our three types. Recall that
as well as a sesquilinear form
b in two
variables, we have a form f in one variable
— either f is defined by f v C b v v , or b is obtained by polarising f — and
we make use of both forms. A subspace of V on which b vanishes identically is
called a totally isotropic subspace (or t.i. subspace), while a subspace on which f
vanishes identically is called a totally singular subspace (or t.s. subspace). Every
t.s. subspace is t.i., but the converse is false. In the case of alternating forms, every
subspace is t.s.! I frequently use the expression t.i. or t.s. subspace, to mean a t.i.
subspace (in the symplectic or unitary case) or a t.s. subspace (in the orthogonal
case).
The classical polar space (or simply the polar space) associated with a vector
space carrying a form is the geometry whose flats are the t.i. or t.s. subspaces (in
the above sense). (Concerning the terminology: the term “polar space” is normally
reserved for a geometry satisfying the axioms of Tits, which we will meet shortly.
But every classical polar space is a polar space, so the terminology here should
cause no confusion.) Note that, if the form is anisotropic, then the only member
of the polar space is the zero subspace. The polar rank of a classical polar space is
6.4. Classical polar spaces
89
the largest vector space rank of any t.i. or t.s. subspace; it is zero if and only if the
form is anisotropic. Where there is no confusion, polar rank will be called simply
rank. (We will soon see that there is no conflict with our earlier definition of polar
rank as the number of hyperbolic lines in the decomposition of the space.) We use
the terms point, line, plane, etc., just as for projective spaces.
We now proceed to derive some properties of polar spaces. Let G be a classical
polar space of polar rank r.
First, we identify the two definitions of polar space rank. We use the expression for V as the direct sum of r hyperbolic lines and an anisotropic subspace
given by Theorem 6.7. Any t.i. or t.s. subspace meets each hyperbolic line in at
most a point, and meets the anisotropic germ in the zero space; so its rank is at
most r. But the span of r t.i. or t.s. points, one chosen from each hyperbolic line,
is a t.i. or t.s. subspace of rank r.
(P1) Any flat, together with the flats it contains, is a projective space of dimension at most r & 1.
This is clear since a subspace of a t.i. or t.s. subspace is itself t.i. or t.s. The next
property is also clear.
(P2) The intersection of any family of flats is a flat.
(P3) If U is a flat of dimension r & 1 and p a point not in U , then the union of the
lines joining p to points of U is a flat W of dimension r & 1; and U U W is a
hyperplane in both U and W .
Proof Let p V w . The function v b v w on the vector space U is linear; let
K be its kernel, a hyperplane in U . Then the line (of the projective space) joining
p to a point q U is t.i. or t.s. if and only if q K; and the union of all such t.i. or
t.s. lines is a t.i. or t.s. space W E K v , such that W U U K, as required.
(P4) There exist two disjoint flats of dimension r & 1.
Proof Use the hyperbolic-anisotropic decomposition again. If L1 P
P
P
W Lr are the
hyperbolic lines, and vi wi are the distinguished spanning vectors in Li , then the
required flats are v1 P
P
P
W vr and w1 !
P
P
! wr .
Next, we specialise to the case r 2. (A polar space of rank 1 is just an
unstructured collection of points.) A polar space of rank 2 consists of points and
lines, and has the following properties. (The first two are immediate consequences
of (P2) and (P3) respectively.)
90
6. Polar spaces
(Q1) Two points lie on at most one line.
(Q2) If L is a line, and p a point not on L, then there is a unique point of L
collinear with p.
(Q3) No point is collinear with all others.
For, by (P4), there exist disjoint lines; and, given any point p, at least one of
these lines does not contain p, and p fails to be collinear with some point of this
line.
A geometry satisfying (Q1), (Q2) and (Q3) is called a generalised quadrangle. Such geometries play much the same rôle in the theory of polar spaces as
projective planes do in the theory of projective spaces. We will return to them
later.
Note that (Q1) holds in a polar space of arbitrary rank.
Another property of polar spaces, which is proved by almost the same argument as (P3), is the following extension of (Q2):
(BS) If L is a line, and p a point not on L, then p is collinear with one or all
points of L.
In a polar space G, for any set S of points, we let S denote the set of points
which are perpendicular to (that is, collinear with) every point of S. It follows
from (BS) that, for any set S, the set S is a (linear) subspace of G (that is, if two
points of S are collinear, then the line joining them lies wholly in S ). Moreover,
for any point x, x is a hyperplane of G (that is, a subspace which meets every
line).
Polar spaces have good inductive properties. Let G be a classical polar space.
There are two natural ways of producing a “smaller” polar space from G:
(a) Take a point x of G, and consider the quotient space x A x, the space whose
points, lines, . . . are the lines, planes, . . . of G containing x.
(b) Take two non-perpendicular points x and y, and consider x y .
In each case, the space constructed is a classical polar space, having the same
germ as G but with polar rank one less than that of G. (Note that, in (b), the
span of x and y in the vector space is a hyperbolic line.) There are more general
versions. For example, if S is a flat of dimension d & 1, then S A S is a polar space
6.4. Classical polar spaces
91
of rank r & d with the same germ as G. We will see below and in the next section
how this inductive process can be used to obtain information about polar spaces.
We investigate just one type in more detail, the so-called hyperbolic quadric
or hyperbolic orthogonal space, the orthogonal space which is a direct sum of
hyperbolic lines (that is, having germ 0). The quadratic form defining this space
can be taken to be x1 x2 x3 x4 P
P
x2r 1 x2r .
3
Theorem 6.11 The maximal flats of a hyperbolic quadric fall into two classes,
with the properties that the intersection of two maximal flats has even codimension
in each if and only if they belong to the same class.
Proof First, note that the result holds when r 1, since then
the quadratic form is
x1 x2 and there are just two singular points, 1 0 P and 0 1 ! . By the inductive
principle, it follows that any flat of dimension r & 2 is contained in exactly two
maximal flats. We take the r & 1 -flats and r & 2 -flats as the vertices
and edges of a graph Γ,
that is, we join two r & 1 -flats if their intersection is an r & 2 -flat. The theorem
will follow if we show that Γ is connected and bipartite, and that the distance
between two vertices of Γ is the codimension of their intersection. Clearly the
codimension of the intersection increases by at most one with every step in the
graph, so it is at most equal to the distance.
We prove equality by induction.
Let U be a r & 1 -flat and K a r & 2 -flat. We claim that the two r & 1 spaces W1 W2 containing K have different distances from U . Factoring out the
/ Then
t.s. subspace U U K and using induction, we may assume that U U K 0.
U U K is a point p, which lies in one but not the other of W1 W2 ; say p W1 . By
induction, the distance from U to W1 is r & 1; so the distance from U to W2 is at
most r, hence equal to r by the remark in the preceding paragraph.
This establishes the claim about the distance. The fact that Γ is bipartite also
follows, since in any non-bipartite graph there exists an edge both of whose vertices have the same distance from some third vertex, and the argument given shows
that this doesn’t happen in Γ.
In particular, the rank 2 hyperbolic quadric consists of two families of lines
forming a grid, as shown in Fig. 6.1. This is the so-called “ruled quadric”, familiar
from models such as wastepaper baskets.
Exercises
1. Prove (BS).
92
6. Polar spaces
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b
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cb
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c
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cb
Y
Z\ X X \ _ Y a [ [ ] ] ^ c ^ ` ` c
c
Z_ a Y
c
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_ Za
c ^
c
a`
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_\ a\ Z X X c Y Y [ c ]
[c ]
a
^
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a
[
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c [
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[c
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[ _X
c `
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c
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[ X
b c ` a a ^ _ ] _ ] \ [ \ Z Y YZ X
cb
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c
b a a _ ` _ \ ^ \ Z] Z ] X [ Y Y
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b
b a a b _ \ Z ` Z X ^ X ] Y ] [Y
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^
a b b _ b \ b Z X `X ` Y ^[ ] [
a
_ b \ Z b X ` Y ]^ [ ]
Y ^
_
b\ Z
` _
\ b Z X X b b Y ` [ [ ]` ^
Z
\
_`
b Y Y [ ] ` ^]
Z b bX
_ ` \
X b
Yb [
Z
\`
X
Z
`
^ \
Figure 6.1: A grid
2. Prove the assertions above about x A x and x y .
3. Show that Theorem 6.11 can be proved
using only properties (P1)–(P4) of
polar spaces together with the fact that an r & 1 -flat lies in exactly two maximal
flats.
6.5
Finite polar spaces
The classification of finite classical polar spaces was achieved by Theorem 6.7.
We subdivide these spaces into six families according to their germ, viz., one
symplectic, two unitary, and three orthogonal. (Forms which differ only by a
scalar factor obviously define the same polar space.) The following table gives
some information about them. In the table, r denotes the polar space rank, n
the vector space rank. The significance of the parameter ε will emerge shortly.
This number, depending only on the germ, carries numerical information about
all spaces in the family. Note that, in the unitary case, the order of the finite field
6.5. Finite polar spaces
93
must be a square.
Type
Symplectic
Unitary
Unitary
Orthogonal
Orthogonal
Orthogonal
n
2r
2r
2r 1
2r
2r 1
2r 2
&
ε
0
1
2
1
2
& 1
0
1
Table 6.1: Finite classical polar spaces
Theorem 6.12 The number of points in a finite polar space of rank 1 is q1 where ε is given in Table 6.1.
ε
1,
Proof Let V be a vector space carrying a form of rank 1 over GF q . Then V
is the orthogonal direct sum of a hyperbolic line L and an anisotropic germ U of
dimension k (say). Let nk be the number of points.
Suppose that k 0. If p is a point of the polar space, then p lies on the hyperplane p ; any other hyperplane containing p is non-degenerate with polar rank 1
and having germ of dimension k & 1. Consider a parallel class of hyperplanes in
the affine space whose hyperplane at infinity is p . Each such hyperplane contains nk 1 & 1 points, and the hyperplane at infinity contains just one, viz., p. So
3
we have
nk & 1 q nk 1 & 1 3
from which it follows that nk 1 n0 & 1 qk . So it is enough to prove the result
for the case k 0, that is, for a hyperbolic line.
In the symplectic case, each of the q 1 projective points on a line is isotropic.
Consider the unitary case. We can take the form to be
P
b x1 y1 x2 y2 P x1 y2 y1 x2 σ
r 2
where
x x x , r q. So the isotropic points satisfy xy yx 0, that is,
Tr xy # 0. How many pairs x y satisfy this? If y 0, then x is arbitrary. If
y 0, then a fixed
of x is in the kernel of the trace map, a set of size q1d 2
1d multiple
2
(since Tr is GF q -linear). So there are
q
q & 1 q1 d
2
1
q & 1 q1 d
2
1
94
6. Polar spaces
vectors, i.e., q1 d 2 1 projective points.
Finally, consider the orthogonal
case. The quadratic form is equivalent to xy,
and has two singular points, 1 0 P and 1 0 P .
Theorem 6.13 In a finite polar space of rank r, there are qr & 1 qr 1 points, of which q2r 3 1 ε are not perpendicular to a given point.
ε
1 PA q &
Proof We let F r be the number of points, and G r the number not perpendicular to a given point. (We do not assume that G r is constant; this constancy
follows from the induction that proves the theorem.) We use the two inductive
principles described at the end of the last section.
Step 1 G r q2 G r & 1 .
Take a point x, and count pairs y z , where y x , z x , and z y . Choosing z first, there are G r choices; then x z is a hyperbolic line, and y is a point in
x z , so there are F r & 1 choices for y. On the other hand, choosing y first, the
lines
through y are the points of the rank r & 1 polar space x A x, and
so there are
F r & 1 of them, with q points different from x on each, giving qF r & 1 choices
for y; then x y and y z are non-perpendicular lines
in y , i.e., points of y A y,
so there are G r & 1 choices for y z , and so qG r & 1 choices for y. thus
G r /e F r & 1 qF r & 1 Oe qG r & 1 .
from which the result follows.
Since G 1 q1 ε , it follows immediately that G r q2r 3
1 ε,
as required.
Step 2 F r 1 qF r & 1 G r .
For this, simply observe (as above) that points perpendicular to x lie on lines
of x A x.
Now it is just a matter of calculation that the function qr & 1 qr ε 1 PA q &
1 satisfies the recurrence of Step 2 and correctly reduces to q1 ε 1 when r 1.
Theorem 6.14 The number of maximal flats in a finite polar space of rank r is
r
∏ 1
f
i 1
qi ε
.
6.5. Finite polar spaces
95
Proof Let H r be this number. Count pairs x U , where U is a maximal flat
and x U . We find that
F r /e H r & 1 H r Oe qr & 1 PA q & 1 so
H r 1 qr ε
H r & 1 Now the result is immediate.
It should now be clear that any reasonable counting question about finite polar
spaces can be answered in terms of q r ε.
7
Axioms for polar spaces
The axiomatisation of polar spaces was begun by Veldkamp, completed by Tits,
and simplified by Buekenhout, Shult, Hanssens, and others. In this chapter, the
analogue of Chapter 3, these results are discussed, and proofs given in some cases
as illustrations. We begin with a discussion of generalised quadrangles, which
play a similar rôle here to that of projective planes in the theory of projective
spaces.
7.1
Generalised quadrangles
We saw the definition of a generalised quadrangle in Section 6.4: it is a rank 2
geometry satisfying the conditions
(Q1) two points lie on at most one line;
(Q2) if the point p is not on the line L, then p is collinear with exactly one point
of L;
(Q3) no point is collinear with all others.
For later use, we represent generalised quadrangles by a diagram with a double
arc, thus:
The axioms (Q1)–(Q3) are self-dual; so the dual of a generalised quadrangle
is also a generalised quadrangle.
Two simple classes of examples are provided by the complete bipartite graphs,
whose points fall into two disjoint sets (with at least two points in each, and whose
97
98
7. Axioms for polar spaces
lines consist of all pairs of points containing one from each set), and their duals,
the grids, some of which we met in Section 6.4. Any generalised quadrangle in
which lines have just two points is a complete bipartite graph, and dually (Exercise 2). We note that any line contains at least two points, and dually: if L were
a singleton line p , then every other point would be collinear with p (by (Q2)),
contradicting (Q3).
Apart from complete bipartite graphs and grids, all generalised quadrangles
have orders:
Theorem 7.1 Let G be a generalised quadrangle in which there is a line with at
least three points and a point on at least three lines. Then the number of points on
a line, and the number of lines through a point, are constants.
Proof First observe that, if lines L1 and L2 are disjoint, then they have the same
cardinality; for collinearity sets up a bijection between the points on L1 and those
on L2 .
Now suppose that L1 and L2 intersect. Let p be a point on neither of these
lines. Then one line through p meets L1 , and one meets L2 , so there is a line
L3 disjoint from both L1 and L2 . It follows that L1 and L2 both have the same
cardinality as L3 .
The other assertion is proved dually.
This proof works in both the finite and the infinite case. If G is finite, we let
s and t be the orders of G; that is, any line has s 1 points and any point lies on
t 1 lines, so that the diagram is
s
t
For the classical polar spaces over GF q , we have s q and t q1 ε , where
ε is given in Table 6.5.1.
From now on, “generalised quadrangle” will be abbreviated to GQ.
The next result summarises some properties of finite GQs.
Theorem 7.2 Let G be a finite GQ with orders s t.
(a) G has s 1 st 1 points and t 1 st 1 lines.
(b) s t divides st s 1 t 1 ;
(c) if s 1, then t s2 ;
7.1. Generalised quadrangles
99
(d) if t 1, then s t 2 .
Proof (a) is proved by elementary counting, like that in Section 6.5. (b) is shown
by an argument involving eigenvalues of matrices, in the spirit of the proof of
the Friendship Theorem outlined in Exercise 2.2.4. (c) is proved by elementary
counting (see Exercise 3), and (d) is dual to (c).
In particular, if s 2, then t 4; and the case t 3 is excluded by (b) above.
So t 1 2 or 4. These three values are realised by the three orthogonal rank 2
polar spaces over GF 2 . We will see, as a special case of a later result, that these
are the only GQs with s 2. However, this result is sufficiently interesting to be
worth another proof which generalises it in a different direction.
Theorem 7.3 Let G be a GQ with orders s 2 and t. Then t there is a unique geometry for each value of t.
1 2 or 4; and
Note the generalisation: t is not assumed to be finite!
Proof Take a point and call it ∞; let Li : i I be the set of lines containing
∞. Number the points other than ∞ on Li as pi0 and
pi1 . Now, for any point q
not collinear with p, there is a function fq : I 0 1 defined by the rule that
the unique point of Li collinear with q is pi fq i . We use the function fq as a label
for q. Let X be the set of points not collinear with ∞. We consider the possible
relationships of points in X . Write q r if q and r are collinear.
1. If q r X satisfy q r, then fq and fr agree in just one position, viz., the
unique index i for which the line Li through ∞ meets the line qr.
2. If q r are not collinear but some point of X is collinear with both, then fq
and fr agree in all but two positions; for all but two values are changed twice, the
remaining two being changed just once.
3. Otherwise, fq fr ; for all the common neighbours of q and r are adjacent
to ∞.
Note too that, for any i I and q X , there is a point r q for which fq and
fr agree only in i, viz., the last point of the line through q meeting Li .
Now suppose (as we may) that I 2, and choose distinct i j k I. Given
q X , choose r
s t X such that fq and fr agree only in i, fr and fs only in j,
and fs and ft only on k. Then clearly fq and ft agree in precisely the three points
i j k, since these values are changed twice and all others three times. By the case
analysis, it follows that I 3 or I 2 3, as required.
The uniqueness also follows from this analysis, with a little more work: we
know enough about the structure of X that the entire geometry can be reconstructed.
100
7. Axioms for polar spaces
Problem. Can there exist a GQ with s finite (s 1) and t infinite?
The proof above shows that there is no such GQ with s 2. It is also known
that there is no GQ with s 3 or s 4 and t infinite, though the proofs are much
harder. (This is due to Kantor and Brouwer for s 3, and Cherlin for s 4.)
Beyond this, nothing is known, though Cherlin’s argument could in principle be
extended to larger values of s.
The GQs with s 2 and t 1 2 have simple descriptions. For s 1 we have
the 3 3 grid. For t 2, take the points to be all the 2-element subsets of a set
of cardinality 6, and the lines to be all partitions of the 6-set into three disjoint
2-subsets. The GQ with order 2 4 is a little harder to describe. The implicit
construction in Theorem 7.3 is one of the simplest — the functions fq are all those
which take the value 1 an even number of times, each such function representing
a unique point. This GQ also arises in classical algebraic geometry, as the Schläfli
configuration of 27 lines in a general cubic surface, lying three at a time in 45
planes.
In the classical polar spaces, the orders s and t are both powers of the same
prime. There are examples where this is not the case — see Exercise 5.
Exercises
1. Prove that the dual of a GQ is a GQ.
2. Prove that a GQ with two points on any line is a complete bipartite graph.
3. Let G be a finite GQ with orders s t, where s 1. Let p1 and p2 be nonadjacent points, and let xn be the number of points p3 adjacent to neither p1 nor
p2 for which there are exactly n common neighbours of p1 p2 and p3 . Show that
∑n n ∑ xn
∑ nxn
s2t st s t s t 1 t 1 1 xn t 1 t t 1 Hence prove that t s2 , with equality if and only if any three pairwise noncollinear points have exactly s 1 common neighbours.
4. In this exercise, we use the terminology of coding theory (as in Section
3.2). Consider the space V of words of length 6 with even weight. This is a vector
space of rank 5 over GF 2 . The “standard inner product” on V is a bilinear form
which is alternating (by the even weight condition); its radical V is spanned by
the unique word of weight 6. Thus, V V is a vector space of rank 4 carrying a
non-degenerate alternating bilinear form. The 15 non-zero vectors of this space
7.2. Diagrams for polar spaces
101
are cosets of V containing a word of weight 2 and the complementary word
of weight 4, and so can be identified with the 2-subsets of a 6-set. Extend this
identification to an isomorphism between the combinatorial description of the GQ
with orders (2, 2) and the rank 2 symplectic polar space over GF 2 .
5. Let q be an even prime power, and let C be a hyperoval in Π PG 2 q ,
a set of q 2 points meeting every line in 0 or 2 points (see Section 4.3). Now
take Π to be the hyperplane at infinity of AG 3 q . Let G be the geometry whose
points are all the points of AG 3 q , and whose lines are all the lines of AG 3 q which meet Π in a point of C. Prove that G is a GQ with orders q 1 q 1 .
6. Construct “free” GQs.
7.2
Diagrams for polar spaces
The inductive properties of polar spaces are exactly what is needed to show
that they are diagram geometries.
Proposition 7.4 A classical polar space of rank n belongs to the diagram
!
with n nodes.
Proof Given a variety U of rank d, the varieties contained in it form a projective
space of dimension d 1, while the varieties containing it are those of the polar
space U U of rank n d; moreover, any variety contained in U is incident with
any variety containing U . Since a rank 2 polar space is a generalised quadrangle,
it follows by induction that residues of varieties are correctly described by the
diagram.
This diagram is commonly referred to as Cn .
By analogy with Section 5.2, it might be thought that any geometry with diagram Cn for n " 3 is a classical polar space. This is false for several reasons,
which we will see at various points. But first, here is one example of a geometry
with diagram C3 which is nothing like a polar space, even though it is highly symmetrical. This geometry was discovered by Arnold Neumaier, and is referred to
as Neumaier’s geometry or the A7 -geometry.
Let X be a set of seven points. The structure of a projective plane PG 2 2 can
be imposed on X in 30 different ways — this number is the index of PGL 3 2 102
7. Axioms for polar spaces
in the symmetric group S7 . Since PGL 3 2 contains no odd permutations, it is
contained in the alternating group A7 with index 15, and so the 30 planes fall into
two orbits of length 15 under A7 . Now we take the points, lines, and planes of
the geometry to be respectively the elements of X , the 3-element subsets of X ,
and one orbit of A7 on PG 2 2 s. Incidence between points and lines, or between
lines and planes, is defined by membership; and every point is incident with every
plane.
It is clear that the residue of a plane is a projective plane PG 2 2 , while
the residue of a line is a digon. Consider the residue of a point x. The lines
incident with
x can be identified with the 2-element subsets of the 6-element set
Y X # x . Given a plane, its three lines containing x partition Y into three
2-sets. It is easy to check that, given such a triple of lines, there are just two
ways to draw the remaining four lines to complete PG 2 2 , and that these two
are related by an odd permutation of X . So our chosen orbit of planes has exactly
one member inducing the given partition of Y , and the planes incident with x can
be identified with all the partitions of Y into three 2-sets. As we saw in Section
7.1, this incidence structure is a generalised quadrangle with order 2 2.
We conclude that the geometry has the diagram
2
2
2
This example shows that, even in a geometry with such a simple diagram,
a variety is not necessarily determined by its point-shadow (all planes have the
same point-shadow!); the intersection of point-shadows of varieties need not be
the point-shadow of a variety, and the points and lines need not form a partial
L
linear space. So the special properties of linear diagrams with all strokes
do not extend. However, classical polar spaces do have these nice properties.
A C3 -geometry in which every point and every plane are incident is called flat.
Neumaier’s geometry is the only known finite example of such a geometry. Some
infinite examples were constructed by Sarah Rees; we now describe these. First,
a re-interpretation of Neumaier’s geometry.
Consider the rank 6 vector space V of all binary words of length 7 having even
weight. On V , we can define a quadratic form by the rule
f v \$
1
2 wt v % mod 2 The bilinear form obtained by polarising f is just the usual dot product, since
wt v w \$ wt v & wt w ' 2v w 7.2. Diagrams for polar spaces
103
It follows that f is non-singular: the only vector orthogonal to V is the all-1 word,
which is not in V . Now the points of X , which index the coordinates, are in oneone correspondence with the seven words of weight 6, which are non-singular
vectors. The lines correspond to the vectors of weight 4, which comprise all the
singular vectors.
We saw in Section 6.4 that the planes on the quadric fall into two families,
such that two planes of the same family meet in a subspace of even codimension
(necessarily a point), while planes of different families meet in a subspace of odd
codimension (the empty set or a line). Now a plane on the quadric contains seven
non-zero singular vectors (of weight 4), any two of which are orthogonal, and so
meet in an even number of points, necessarily 2. The complements of these 4-sets
form seven 3-sets, any two meeting in one point, so forming a projective plane
PG 2 2 . It is readily checked that the two classes of planes correspond exactly
to the two orbits of A7 we described earlier. So the points, lines and planes of
Neumaier’s geometry can be identified with a special set of seven non-singular
points, the singular points, and one family of planes on the quadric. Incidence
between the non-singular and the singular points is defined by orthogonality.
Now we reverse the procedure. We start with a hyperbolic quadric Q in
PG 5 F , that is, a quadric of rank 3 with germ zero. A set S of non-singular
points is called an exterior set if it has the property that, given any line L of
Q, a unique point of S is orthogonal to L. Now consider the geometry G whose
POINTS, LINES and PLANES are the points of S, the points of Q, and one family
of planes on Q; incidence between POINTS and LINES is defined by orthogonality, that between LINES and PLANES is incidence in the polar space, and every
POINT is incident with every PLANE.
Such a geometry belongs to the diagram C3 . For the residue of a PLANE Π is
a projective plane, naturally the dual of Π. (The correspondence between points
of S and lines of Π is bijective; for, given x S, x cannot contain Π, since a polar
space in PG 4 q cannot have rank 3, and so it meets Π in a line.) The residue of
a POINT x is the polar space x , which as we’ve seen is rank 2, and so a GQ. And
of course the POINTS and PLANES incident with a LINE form a digon.
Thas showed that no further finite examples can be constructed in this way:
Theorem 7.5 There is no exterior set for the hyperbolic quadric in PG 5 q for
q 2.
However, Rees (who first described this construction) observed that there are
infinite examples. Consider the case where F )( ; let the form be x1 x2 x3 x4 104
7. Axioms for polar spaces
x5 x6 . Now the space of rank 3 spanned by 1 1 0 0 0 0 , 0 0 1 1 0 0 and
0 0 0 0 1 1 is positive definite, and so is disjoint from the quadric; the points
spanned by vectors in this space form an exterior set.
Now we turn to hyperbolic quadrics in general. As we saw in Section 6.4, the
maximal t.s. subspaces on such a quadric Q of rank n can be partitioned into two
families, so that a flat of dimension n 2 lies in a unique member of each family.
We construct a new geometry by letting these flats be varieties of different types.
Now there is no need to retain the flats of dimension n 2, since such a flat is the
intersection of the two maximal flats containing it.
Theorem 7.6 Let Q be a hyperbolic quadric of rank n " 3. Let G be the geometry
whose flats are the t.s. subspaces of dimension different from n 2, where the
two families of flats of dimension n 1 are assigned different types. Incidence
between flats, at least one of which has dimension less than n 1, is as usual;
while n 1 -flats of different types are incident if they intersect in an n 2 -flat.
Then the geometry has diagram
!
(n nodes).
Proof We need only check the residue of a flat of dimension n 3: the rest
follows by induction, as in Proposition 7.4. Such a flat cannot be the intersection
of two n 1 -flats of different types; so any two such flats of different types
containing it are incident.
This diagram is denoted by Dn . The result holds also for n 2, provided that
we interpret D2 as two unconnected nodes — the quadric has two families of lines,
each line of one family meeting each line of the other.
Exercises
1. Prove that the line joining two points of an exterior set to the quadric Q is
disjoint from Q.
2. Prove that an exterior set to a quadric in PG 5 q must have q2 q 1
points.
3. Show that the plane constructed in Rees’ example is an exterior set.
7.3. Tits and Buekenhout–Shult
7.3
105
Tits and Buekenhout–Shult
We now begin working towards the axiomatisation of polar spaces. This major
result of Tits (building on earlier work of Veldkamp) will not be proved completely
here, but the next four sections should give some impression of how the proof
works.
Tits’ theorem characterises a class of spaces which almost coincides with the
classical polar spaces of rank at least 3. There are a few additional examples
of rank 3, some of which will be described later. I will use the term abstract
polar space for a geometry satisfying the axioms. In fact, Tits’ axioms describe
all subspaces of arbitrary dimension; an alternative axiom system, proposed by
Buekenhout and Shult, involves only points and lines (in the spirit of the Veblen–
Young axioms for projective spaces). In this section, I show the equivalence of
these axiom systems.
Temporarily, then, an abstract polar space of type T is a geometry satisfying
the conditions (P1)–(P4) of Section 6.4, repeated here for convenience.
(P1) Any flat, together with the flats it contains, is a projective space of dimension at most r 1.
(P2) The intersection of any family of flats is a flat.
(P3) If U is a flat of dimension r 1 and p a point not in U , then the union of the
lines joining p to points of U is a flat W of dimension r 1; and U * W is a
hyperplane in both U and W .
(P4) There exist two disjoint flats of dimension r 1.
An abstract polar space of type BS is a geometry of points and lines satisfying
the following conditions. In these axioms, a subspace is a set S of points with the
property that if a line L contains two points of S, then L + S; a singular subspace
is a subspace, any two of whose points are collinear.
(BS1) Any line contains at least three points.
(BS2) No point is collinear with all others.
(BS3) Any chain of singular subspaces is of finite length.
(BS4) If the point p is not on the line L, then p is collinear with one or all points
of L.
106
7. Axioms for polar spaces
(Note that (BS4) is our earlier (BS), and is the key condition here.)
Theorem 7.7 (a) The points and lines of an abstract polar space of type T form
an abstract polar space of type BS.
(b) The singular subspaces of an abstract polar space of type BS form an abstract
polar space of type T.
Proof (a) It is an easy deduction from (P1)–(P4) that any subspace is contained
in a subspace of dimension n 1. For let U be a subspace, and W a subspace
of dimension n 1 for which U * W has dimension as large as possible; if p U # W , then (P3) gives a subspace of dimension n 1 containing p and U * W ,
contradicting maximality.
Now, if L is a line and p a point not on L, let W be a subspace of dimension
n 1 containing L. If p W , then p is collinear with every point of L; otherwise,
the neighbours of p in W form a hyperplane, meeting L in one or all of its points.
Thus, (BS4) holds. The other conditions are clear.
(b) Now let G be an abstract polar space of type BS. Call two points adjacent
if they are collinear; this gives the point set a graph structure. Every maximal
clique in the graph is a subspace. For let S be a maximal clique, and p q S; let
L be a line containing p and q. Any point of S # L is collinear with p and q, and so
with every point of L; thus S , L is a clique, and by maximality, L + S.
If p - S (where S is a maximal clique), then the set of neighbours of p in S is a
hyperplane. Every point q S lies outside such a hyperplane; for, by (BS2), there
is a point p not adjacent to q. As we saw in Section 3.1, if every line has size 3,
then this implies that S is a projective space; but this deduction cannot be made in
general. However, in the present situation, Buekenhout and Shult are able to show
that S is indeed a projective space. (In particular, this implies that two points lie on
at most one line. For the union of two lines through two common points is a clique
by (BS4), and so would be contained in a maximal clique. However, Buekenhout
and Shult have to show that two points lie on at most one line before they know
that the subspaces are projective spaces; the proof is surprisingly tricky.)
Any singular subspace lies in some maximal clique, and so is itself a projective
space. Thus (P1) holds; and the remaining axioms can now be verified.
We will now simplify the terminology by using the term “abstract polar space”
equally for either type.
The induction principles we used in classical polar spaces work in almost the
same way in abstract polar spaces.
7.4. Recognising hyperbolic quadrics
107
Proposition 7.8 Let U be a d 1 -dimensional subspace of an abstract polar
space of rank n. Then the subspaces containing U form an abstract polar space
of rank n d.
Exercise
1. Show directly that, in an abstract polar space of type BS having three points
on any line, any two points lie on at most one line, and singular subspaces are
projective.
7.4
Recognising hyperbolic quadrics
There are two special cases where the proof of the characterisation of polar spaces is substantially easier, namely, hyperbolic quadrics and quadrics over
GF 2 ; they will be treated in this section and the next.
In the case of a hyperbolic quadric, we bypass the need to reconstruct the
quadric by simply showing that there is a unique example of each rank over any
field. First, we observe that the partition of the maximal subspaces into two types
follows directly from the axioms; properties of the actual model are not required.
We begin with a general result on abstract polar spaces.
An abstract polar space G can be regarded as a point-line geometry, as we’ve
seen. Sometimes it is useful to consider a “dual” situation, defining a geometry G .
whose POINTs are the maximal subspaces of G and whose LINEs are the nextto-maximal subspaces, incidence being reversed inclusion. We call this geometry
a dual polar space. In a dual polar space, we define the distance between two
POINTs to be the number of LINEs on a shortest path joining them.
Proposition 7.9 Let G . be a dual polar space.
(a) The distance between two POINTs is the codimension of their intersection.
(b) Given a POINT p and a LINE L, there is a unique POINT of L nearest to p.
Proof Let U1 , U2 be maximal subspaces. By the inductive principle (Proposi/ (It is clear that any path from U1 to
tion 7.8), we may assume that U1 * U2 0.
U2 , in which not all terms contain U1 * U2 , must have length strictly greater than
the codimension of U1 * U2 ; so, once the result is proved in the quotient, no such
path can be minimal.)
108
7. Axioms for polar spaces
Now each point of U1 is collinear (in G) with a hyperplane in U2 , and vice
versa; so, given any hyperplane H in U2 , there is a unique point of U1 adjacent to
H, and hence (by (P3)) a unique maximal subspace containing H and meeting U1 .
The result follows.
(b) Let U be a maximal subspace and W a subspace of rank one less than
/ Now there is a unique point
maximal. As before, we may assume that U * W 0.
p U collinear with all points of W . Then / W p 0 is the uniue POINT on the LINE
W nearest to the POINT U .
Proposition 7.10 Let G be an abstract polar space of rank n, in which any n 2 -dimensional subspace is contained in exactly two maximal subspaces. Then
the maximal subspaces fall into two families, the intersection of two subspaces
having even codimension in each if and only if the subspaces belong to the same
family.
Proof The associated dual polar space is a graph. By Proposition 7.9(b), the
graph is bipartite, since if an odd circuit exists, then there is one of minimal length,
and both vertices on any edge are then equidistant from the opposite vertex in the
cycle.
Now, in any abstract polar space of rank n " 4, in which lines contain at least
three points, any maximal subspace is isomorphic to PG n 1 F for some skew
field F. Now an easy connectedness argument shows that the same field F coordinatises every maximal subspace.
Theorem 7.11 Let G be an abstract polar space of rank n " 4, in which each nextto-maximal subspace is contained in exactly two maximal subspaces. Assume that
some maximal subspace is isomorphic to PG n 1 F . Then F is commutative,
and G is isomorphic to the hyperbolic quadric of rank n over F.
Proof It is enough to show that F is commutative and that n and F uniquely determine the geometry, since the hyperbolic quadric clearly has the required property.
Rather than prove F commutative, I will show merely that it is isomorphic
to its opposite. It suffices to show this when n 4. Take two maximal subspaces meeting in a plane Π, and a point p Π. By the FTPG, both maximal
subspaces are isomorphic to PG 3 F . Now consider the residue of p. This is a
projective space, in which there is a plane isomorphic to PG 2 F 12 , and a point
residue isomorphic to PG 2 F . Hence F F 1 . The stronger statement that F is
commutative is shown by Tits. He observes that the quotient of p has a polarity
7.5. Recognising quadrics over GF 2 109
interchanging a point and a plane incident with it, and fixing every line incident
with both; and this can only happen in a projective 3-space over a commutative
field.
Let U1 and U2 be disjoint maximal subspaces. Note that they have the same
type if n is even, opposite types if n is odd. Let p be any point in neither subspace.
Then for i 1 2, there is a unique maximal subspace Wi containing p and meeting
Ui in a hyperplane. Then Wi has the opposite type to Ui , so W1 and W2 have the
same type if n is even, opposite types if n is odd. Thus, their intersection has
codimension congruent to n mod 2. Since p W1 * W2 , the intersection is at least
a line. But their distance in the dual polar space is at least n 2, since U1 and U2
have distance n; so W1 * W2 is a line L. Clearly L meets both U1 and U2 .
Each point of U1 is adjacent to a hyperplane of U2, and vice versa; so U1
and U2 are naturally duals. Now the lines joining points of U1 and U2 are easily
described, and it is not hard to show that the whole geometry is determined.
7.5
Recognising quadrics over GF 3 2 4
In this section, we determine the abstract polar spaces with three points on
every line. Since we are given information only about points and lines, the BS
approach is the natural one. The result here was first found by Shult (assuming
a constant number of lines per point) and Seidel (in general), and was a crucial
precursor of the Buekenhout–Shult Theorem (Theorem 7.7). Shult and Seidel
proved the theorem by induction on the rank: a rank 2 polar space is a generalised
quadrangle, and the classification in this case is Theorem 7.3. The elegant direct
argument given here is due to Jonathan Hall.
Let G be an abstract polar space with three points per line. We have already
seen that the facts that two points lie on at most one line, and that maximal singular
subspaces are projective spaces, are proved more easily under this hypothesis than
in general. But here is a direct proof of the first assertion. Suppose that the points
a b y . Then y is collinear with a and b, and
a and b lie on two lines a b x and
so also with x; so there is a line x y
z for some z, and both a and b are joined
to z. Any further point is joined to both or neither x and y, and so is joined to z,
contradicting (BS2).
Define a graph Γ whose vertices are the points, two vertices being adjacent if
they are collinear. The graph has the following property:
(T) every edge x y lies in a triangle
x y
z with the property that any further
point is joined to one or all of x y
z .
110
7. Axioms for polar spaces
This is called the triangle property. Shult and Seidel phrased their result as the
determination of finite graphs with the triangle property. (The argument just given
shows that, in a graph with the triangle property in which no vertex is adjacent to
all others, there is a unique triangle with the property specified by (T) containing
any edge. Thus, the graph and the polar space determine each other.) The proof
given below is not the original argument of Shult and Seidel, which used induction, but is a direct argument due to Jonathan Hall (having the added feature that
it works equally well for infinite-dimensional spaces).
Theorem 7.12 An abstract polar space in which each line contains three points
is a quadric over GF 2 .
Proof As noted above, we may assume instead that we have a graph Γ with the
triangle property (T), having at least one edge, and having no vertex adjacent to
all others. Let X be the vertex set of the graph Γ, and let F GF 2 . We begin
with the vector space V̂ of all functions from X to F which are zero everywhere
except on a finite set, with pointwise operations. (If X is finite, then V is just the
space F X of all functions
from X to F.) Let x̂ V̂ be the characteristic function of
the singleton set x . The functions x̂, for x X , form a basis for V̂ . We define a
bilinear form b̂ on V̂ by setting
b̂ x̂ ŷ \$65
0
1
if x y or x is joined to y,
otherwise,
and extending linearly, and a quadratic form fˆ by setting fˆ x̂ 7 0 for all x X
and extending to V̂ by the rule
fˆ v w !
fˆ v &
fˆ w b̂ v w Note that both b̂ and fˆ are well-defined.
Let R be the radical of fˆ; that is, R is the subspace
v V̂ : fˆ v 8 0 b̂ v w 8 0 for all w V̂ 9
and set V V̂ R. Then b̂ and fˆ induce bilinear and quadratic forms b f on V : for
example, we have f v R \$ fˆ v (and this is well-defined, that is, independent
of the choice of coset representative). Now let x̄ x̂ R V .
We claim that the embedding x : x̄ has the required properties; in other words,
it is one-to-one; its image is the quadric defined by f ; and two vertices are adjacent
if and only if the corresponding points of the quadric are orthogonal. We proceed
in a series of steps.
7.5. Recognising quadrics over GF 2 111
Step 1 Let x y
z be a special triangle, as in the statement of the triangle property (T). Then x̄ ȳ z̄ 0.
It is required to show that r x̂ ŷ ẑ R. We have
b̂ r
v̂ \$ b̂ x̂ v̂ b̂ ŷ v̂ b̂ ẑ v̂ \$ 0
for all v X , by the triangle property; and
fˆ r \$
fˆ x̂ &
fˆ ŷ &
fˆ ẑ b̂ x̂ ŷ b̂ ŷ ẑ b̂ ẑ x̂ 8 0
by definition.
Step 2 The map x : x̄ is one-to-one on X .
Suppose that x̄ ȳ. Then r x̂ ŷ R. Hence b̂ x̂ ŷ ! 0, and so x is joined
to y. Let z be the third vertex of the special triangle containing x and y. Then
ẑ x̂ ŷ R by Step 1, and so z is joined to all other points of X , contrary to
assumption.
Step 3 Any quadrangle is contained in a 3 3 grid.
Let x y
z w be a quadrangle. Letting x y x̄ ȳ, etc., we see that x y
is not joined to z or w, and hence is joined to z w. Similarly, y z is joined
to w x; and the third point in the special triangle through each of these pairs is
x y z w, completing the grid. (See Fig. 7.1.)
Step 4 For any v V , write v ∑i ; I x̄i , where xi X , and the number m < I of summands is minimal (for the given v). Then
(a) m 3;
(b) the points xi are pairwise non-adjacent.
This is the crucial step, and needs four sub-stages.
Substep 4.1 Assertion (b) is true.
If xi x j , we could replace xi x j by the third point xk of the special triangle,
and obtain a shorter expression.
112
7. Axioms for polar spaces
z w
w
z
x y z w
w x
x y
x
y z
y
Figure 7.1: A grid
Substep 4.2 If L is a line on x1 , and y a point of L which is adjacent to x2 ,
then y xi for all i I.
If not, let L x1 y
z , and suppose that xi z. Then xi is joined to the third
point w of the line x2 y. Let u be the third point on xi w. Then z̄ p̄1 ū x̄i x̄2 ,
and we can replace x̄1 x̄2 x̄i by the shorter expression z̄ ū.
Substep 4.3 There are two points y
z joined to all xi .
Each line through x1 contains a point with this property, by Substep 4.2. It is
easily seen that if x1 lies on a unique line, then one of the points on this line is
adjacent to all others, contrary to assumption.
Substep 4.4 m 3.
Suppose not. Considering the quadrangles x1 y
x2 z and x3 y
x4 z , we
find (by Step 3) points a and b with
x̄1 ȳ x̄2 z̄ ā x̄3 ȳ x̄4 z̄ b̄ But then x̄1 x̄2 x̄3 x̄4 ā b̄, a shorter expression.
Step 5 If v V , v - 0, and f v \$ 0, then v x̄ for some x X .
7.6. The general case
113
If not then, by Step 4, either v x̄ ȳ, or v x̄ ȳ z̄, where points x y (and
z) are (pairwise) non-adjacent. In the second case,
f v 8
f x̄ &
f ȳ =
f z̄ = b x̄ ȳ = b ȳ z̄ & b z̄ x̄ 8 0 0 0 1 1 1 1 The other case is similar but easier.
Step 6 x y if and only if b x̄ ȳ \$ 0.
This is true by definition.
7.6
The general case
A weak form of the general classification of polar spaces, by Veldkamp and
Tits, can be stated as follows.
Theorem 7.13 A polar space of type T having finite rank n " 4 is either classical,
or defined by a pseudoquadratic form on a vector space over a division ring of
characteristic 2.
I will not attempt to outline the proof of this theorem, but merely make some
remarks, including a “definition” of a pseudoquadratic form.
Let V be a vector space over a skew field F of characteristic 2, and σ an
anti-automorphism
of F satisfying σ2 1. Let K0 be the additive subgroup
x xσ of F, and K .> K K0 . A function f : V K . is called a pseudoquadratic form relative to σ if there is a σ-sesquilinear form g such that f v ?
g v v mod K0 . Equivalently, f polarises to a σ-Hermitian form f satisfying
A@ v V CB c F f v v ? c cσ , that is, a trace-valued form. The function
f defines a polar space, consisting of the subspaces of V on which f vanishes
(mod K0 ). If K0 is equal to the fixed field of σ, then the same polar space is defined by the Hermitian form g; so we may assume that this is not the case in the
second conclusion of Theorem 7.13. For further discussion, see Tits [S].
Tits’ result is actually better than indicated: all polar spaces of rank n " 3 are
classified. There are two types of polar spaces of rank 3 which are not covered
by Theorem 7.13. The first exists over any non-commutative field, and will be
described in the first section of Chapter 8. The other is remarkable in consisting of
the only polar spaces whose planes are non-Desarguesian. This type is constructed
by Tits from the algebraic groups of type E6 , and again I refer to Tits for the
construction, which requires detailed knowledge of these algebraic groups. The
114
7. Axioms for polar spaces
planes actually satisfy a weakening of Desargues’ theorem known as the Moufang
condition, and can be “coordinatised” by certain alternative division rings which
generalise the Cayley numbers or octonions.
Of course, the determination of polar spaces of rank 2 (GQs) is a hopeless task!
Nevertheless, it is possible to formulate the Moufang condition for generalised
quadrangles; and all GQs satisfying the Moufang condition have been determined
(by Fong and Seitz in the finite case, Tits and Weiss in general.) This effectively
completes the analogy with coordinatisation theorems for projective spaces.
The other geometric achievement of Tits in the 1974 lecture notes is the analogue of the Fundamental Theorem of Projective Geometry:
Theorem 7.14 Any isomorphism between classical polar spaces of rank at least
2, which are not of symplectic or orthogonal type in characteristic 2, is induced
by a semilinear transformation of the underlying vector spaces.
The reason for the exception will be seen in Section 8.4. As in Section 1.3,
this result shows that the automorphism groups of classical polar spaces consist of
semilinear transformations modulo scalars. These groups, with some exceptions
of small rank, have “large” simple subgroups, just as happened for the automorphism groups of projective spaces in Section 4.6. These groups are the classical
groups, and are named after their polar spaces: symplectic, orthogonal and unitary groups. For details, see the classic accounts: Dickson [K], Dieudonné [L],
and Artin [B], or for more recent accounts Taylor [R], Cameron . In the symplectic or unitary case, the classical group consists of all the linear transformations
of determinant 1 preserving the form defining the geometry, modulo scalars. In
the orthogonal case, it is sometimes necessary to pass to a subgroup of index 2.
(For example, if the polar space is a hyperbolic quadric in characteristic 2, take
the subgroup fixing the two families of maximal t.s. subspaces.)
8
The Klein quadric and triality
Low-dimensional hyperbolic quadrics possess a remarkably rich structure; the
Klein quadric in 5-space encodes a projective 3-space, and the triality quadric in
7-space possesses an unexpected threefold symmetry. The contents of this chapter
can be predicted from the diagrams of these geometries, since D3 is isomorphic to
A3 , and D4 has an automorphism of order 3.
8.1
The Pfaffian
The determinant of a skew-symmetric matrix is a square. This can be seen in
small cases by direct calculation:
0
a12
det a13
a14
Theorem 8.1
zero.
det
a12
0
a12
0 a12
0
a23
a24
a13
a23
0
a34
a14 a24
a34 0
a212 a12 a34 a13 a24 a14 a23 2 (a) The determinant of a skew-symmetric matrix of odd size is
(b) There is a unique polynomial Pf A in the indeterminates ai j for 1 i 2n, having the properties
j
(i) if A is a skew-symmetric 2n 2n matrix with i j entry ai j for 1 i j 2n, then
Pf A 2;
det A 115
116
8. The Klein quadric and triality
(ii) Pf A contains the term a12 a34 a2n 1 2n
with coefficient 1.
Proof We begin by observing that, if A is a skew-symmetric matrix, then the
form B defined by
xAy B x y
is an alternating bilinear form. Moreover, B is non-degenerate if and only if A is
non-singular: for xAy
0 for all y if and only if xA 0. We know that there is
no non-degenerate alternating bilinear form on a space of odd dimension; so (a)
is proved.
0, whereas if A is nonWe know also that, if A is singular, then det A singular, then there exists an invertible matrix P such that
PAP diag
0
1
1 0 0
1
1
0 so that det A det P 2 . Thus, det A is a square in either case.
Now regard ai j as being indeterminates over the field F; that is, let K F ai j :
1 i j 2n be the field of fractions of the polynomial ring in n 2n 1 vari
ables over F. If A is the skew-symmetric matrix with entries ai j for 1 i j 2n, then as we have seen, det A is a square in K. It is actually the square
of a polynomial. (For the polynomial ring is a unique factorisation domain; if
f g 2 , where f and g are polynomials with no common factor, then
det A det A g2 f 2 , and so f 2 divides det A ; this implies that g is a unit.) Now det A contains a term
a212 a234 a22n 1 2n
corresponding to the permutation
1 2 3 4
2n 1 2n and so by choice of sign in the square root we may assume that (ii)(b) holds.
Clearly the polynomial Pf A is uniquely determined.
The result for arbitrary skew-symmetric matrices is now obtained by specialisation (that is, substituting values from F for the indeterminates ai j ).
Exercises
1. A one-factor on the set 1 2 2n is a partition F of this set into n subsets
of size 2. We represent each 2-set i j by the ordered pair i j with i j. The
crossing number χ F of the one-factor F is the number of pairs i j k l of
sets in F for which i k j l.
8.2. The Klein correspondence
(a) Let (b) Let A
n
117
be the set of one-factors on the set 1 2 2n . What is !
n?
ai j be a skew-symmetric matrix of order 2n. Prove that
Pf A ∑
F "\$#
n
1 χ % F &
∏
% i ' j &(" F
ai j 2. Show that, if A is a skew-symmetric matrix and P any invertible matrix,
then
Pf PAP det P ) Pf A Hint: We have det PAP det P 2 det A , and taking the square root shows that
Pf PAP det P Pf A ; it is enough to justify the positive sign. Show that it
+*
suffices to consider the ‘standard’ skew-symmetric matrix
,
A
diag
0
1
1 0 0
1
1
0 In this case, show that the 2n 1 2n entry in PAP contains the term p2n 1 2n 1 p2n 2n ,
so that Pf PAP contains the diagonal entry of det P with sign 1.
3. Show that any linear transformation of a vector space fixing a symplectic
form (a non-degenerate alternating bilinear form) has determinant 1.
8.2
The Klein correspondence
We begin by describing an abstract polar space which appears not to be of
classical type. Let F be a skew field, and consider the geometry - defined from
PG 3 F as follows:
.
.
the POINTs of - are the lines of PG 3 F ;
.
the LINEs of - are the plane pencils (incident point-plane pairs);
the PLANEs of - are of two types: the points, and the planes.
A POINT and LINE are incident if the line belongs to the plane pencil (i.e., is
incident with both the point and the plane). A LINE and PLANE are incident
if the point or plane is one of the elements of the incident pair; and incidence
between a POINT and a PLANE is the usual incidence in PG 3 F .
If a PLANE is a plane Π, then the POINTs and LINEs of this PLANE correspond to the lines and points of Π; so the residue of the plane is isomorphic to the
118
8. The Klein quadric and triality
dual of Π, namely, PG 2 F / . On the other hand, if a PLANE is a point p, then
the POINTs and LINEs of this PLANE are the lines and planes through p, so its
residue is the residue of p in PG 3 F , namely PG 2 F . Thus (PS1) holds. (Note
that, if F is not isomorphic to its opposite, then the space contains non-isomorphic
planes, something which cannot happen in a classical polar space.)
Axiom (PS2) is clear. Consider (PS3). Suppose that the PLANE in question
is a plane Π, and the POINT not incident with it is a line L. Then L 0 Π is a point
p; the set of POINTs of Π collinear with L is the plane pencil defined by p and Π
(which is a LINE), and the union of the LINEs joining them to L consists of all
lines through p (which is a PLANE), as required. The other case is dual.
Finally, if the point p and plane Π are non-incident, then the PLANEs they
define are disjoint, proving (PS4).
Note that any LINE is incident with just two PLANEs, one of each type; so, if
the polar space is classical, it must be a hyperbolic quadric in PG 5 F . We now
show that, if F is commutative, it is indeed this quadric in disguise! (For noncommutative fields, this is one of the exceptional rank 3 polar spaces mentioned
in Section 7.6.)
The skew-symmetric matrices of order 4 over F form a vector space of rank
6, with x12 x34 as coordinates. The Pfaffian is a quadratic form on this vector
space, which vanishes precisely on the singular matrices. So, projectively, the
singular matrices form a quadric 1 in PG 5 F , the so-called Klein quadric. From
the form of the Pfaffian, we see that this quadric is hyperbolic — but in fact this
will become clear geometrically.
Any skew-symmetric matrix has even rank. In our case, a non-zero singular
skew-symmetric matrix A has rank 2, and so can be written in the form
A
X v w : v w w v
for some vectors v w. Replacing these two vectors by linear combinations αv βw and γv δw multiplies A by a factor αδ βγ (which is just the determinant of
the transformation). So we have a map from the line of PG 3 F spanned by v and
w to the point of the Klein quadric spanned by X v w . This map is a bijection:
we have seen that it is onto, and the matrix determines the line as its row space.
This bijection has the properties predicted by our abstract treatment. Most
important,
two points of the Klein quadric are perpendicular if and only if the
corresponding lines intersect.
8.2. The Klein correspondence
119
To prove this, note that two points are perpendicular if and only if the line
joining them lies in 1 . Now, if two lines intersect, we can take them to be 2 u v 3
and 2 u w 3 ; and we have
α u v v u 4 β u w w u u αv βw αv βw u so the line joining the corresponding points lies in the quadric. Conversely, if two
lines are skew, then they are 2 v1 v2 3 and 2 v3 v4 3 , where v1 5 v4 is a basis;
then the matrix
v1 v2 v2 v1 v3 v4 v4 v3
has rank 4, and is a point on the line not on 1 .
Hence the planes on the quadric correspond to maximal families of pairwise
intersecting lines, of which there are two types: all lines through a fixed point;
and all lines in a fixed plane. Moreover, the argument in the preceding paragraph
shows that lines on 1 do indeed correspond to plane pencils of lines in PG 3 F .
This completes the identification.
Exercise
1. This exercise gives the promised identification of PSL 4 2 with the alter
nating group A8 .
Let V be the vector space of rank 6 over GF 2 consisting of the binary words
of length 8 having even weight modulo the subspace Z consisting of the all-zero
and all-1 words. Show that the function
f v Z
1
2 wt v
mod 2 is well-defined and is a quadratic form of rank 3 on V , whose zeros form the
Klein quadric 1 . Show that the symmetric group S8 interchanges the two families
of planes on 1 , the subgroup fixing the two families being the alternating group
A8 .
Use the Klein correspondence to show that A8 is embedded as a subgroup
of PGL 4 2 PSL 4 2 . By calculating the orders of these groups, show that
equality holds.
Remark The isomorphism between PSL 4 2 and A8 can be used to give a solu
tion to Kirkman’s schoolgirl problem. This problem asks for a schedule for fifteen
schoolgirls to walk in five groups of three every day for seven days, subject to the
120
8. The Klein quadric and triality
requirement that any two girls walk together in a group exactly once during the
week.
The 7 5 groups of girls are thus the blocks of a 2- 15 3 1 design. We will
take this design to consist of the points and lines of PG 3 2 . The problem is then
to find a ‘parallelism’ or ‘resolution’, a partition of the lines into seven ‘parallel
classes’ each consisting of five pairwise disjoint lines.
One way to find a parallel class is to consider the underlying vector space
V 4 2 as a vector space of rank 2 over GF 4 . The five ‘points’ or rank 1
subspaces over GF 4 become five pairwise disjoint lines when we restrict the
scalars to GF 2 . Scalar multiplication by a primitive element of GF 4 is an
automorphism of order 3, fixing all five lines, and commuting with a subgroup
SL 2 4 76 A5 . Moreover, if two such automorphisms oof order 3 have a com
mon fixed line, then they generate a 2 3 -group, since the stabiliser of a line in
GL 4 2 is a 2 3 -group.
Now, in A8 , an element of order 3 commuting with a subgroup isomorphic to
A5 is necessarily a 3-cycle. Two 3-cycles generate a 2 3 -group if and only if
their supports intersect in 0 or 2 points. So we require a set of seven 3-subsets of
1 5 8 , any two of which meet in one point. The lines of PG 2 2 (omitting
one point) have this property.
8.3
Some dualities
We have interpreted points of the Klein quadric in PG 3 F . What about the
points off the quadric?
Theorem 8.2 There is a bijection from the set of points p outside the Klein quadric
1 to symplectic structures on PG 3 F , with the property that a point of 1 per
pendicular to p translates under the Klein correspondence to a totally isotropic
line for the symplectic geometry.
9
Proof A point p 8 1 is represented by a skew-symmetric matrix A which has
non-zero Pfaffian (and hence is invertible), up to a scalar multiple. The matrix
defines a symplectic form b, by the rule
b v w
vAw We must show that a line is t.i. with respect to this form if and only if the corresponding point of 1 is perpendicular to p.
8.3. Some dualities
121
Let A be a non-singular skew-symmetric 4 4 matrix over a field F. By direct
calculation, we show that the following assertions are equivalent, for any vectors
9
v w F 4:
(a) X v w v w w v is orthogonal to A, with respect to the bilinear form
obtained by polarising the quadratic form Q X Pf X ;
(b) v and w are orthogonal with respect to the symplectic form with matrix A† ,
that is, vA† w 0.
Here the matrices A and A† are given by
A
0
a12
a13
a14
a12
0
a23
a24
a14 a24
a34 0
a13
a23
0
a34
A†
0
a34
a24
a23
a34
0
a14
a13
a24
a14
0
a12
a23 a13
a12 0
Note that, if A is the matrix of the standard symplectic form, then so is A† . In
general, the map taking the point outside the quadric spanned by A to the symplectic form with matrix A† is the one asserted in the theorem.
Let - 1 be the symplectic GQ over F, and - 2 the orthogonal GQ associated
9
with the quadric v :;0<1 , where 1 is the Klein quadric and 2 v 38 1 . (Note that
any non-singular quadratic form of rank 2 in 5 variables is equivalent to αx20 x1 x2 x3 x4 for some α 8 0; so any two such forms are equivalent up to scalar
multiple, and define the same GQ.) We have defined a map from points of - 2 to
lines of - 1 . Given any point p of - 1 , the lines of - 1 containing p form a plane
pencil in PG 3 F , and so translate into a line of - 2 . Thus we have shown:
Theorem 8.3 For any field F, the symplectic GQ in PG 3 F and the orthogonal
GQ in PG 4 F are dual.
Now let F be a field which has a Galois extension K of degree 2 and σ the
Galois automorphism of K over F. With the extension K F we can associate two
GQs:
->1= : the unitary GQ in PG 3 K , defined by the Hermitian form
x1 yσ2 x2 yσ1 x3 yσ4 x4 yσ3 ;
->2= : the orthogonal GQ in PG 5 F defined by the quadratic form
x1 x2 x3 x4 αx25 βx5 x6 γx26 where αx2 βx γ is an irreducible quadratic over F which splits in K.
122
8. The Klein quadric and triality
Theorem 8.4 The two GQs ->1= and ->2= defined above are dual.
Proof This is proved by “twisting the Klein correspondence”. In outline, we take
the Klein correspondence over K, and change coordinates on the quadric so that
restriction of scalars to F gives the geometry - 2= , rather than the Klein quadric
over F; then show that the corresponding set of lines in PG 3 K are those which
are totally isotropic with respect to a Hermitian form.
Exercises
1. Prove the assertion about A and A† in the proof of Theorem 8.2.
Let 1 be a hyperbolic quadric of rank n. If v is a non-singular vector, then the
quadric v: 0?1
has the property
A@
.
@
meets every maximal subspace E of 1 in a hyperplane of E.
We call a set satisfying this condition special. The point of the next three ex@
ercises is to investigate whether special sets are necessarily quadrics of the form
v : 0?1 .
2. Consider the case n 2. Let the rank 4 vector space be the space of all
2 2 matrices over F, and let the quadratic form be the determinant.
(a) Show that the map
2 X 3CBD
Ker X Im X induces a bijection between the point set of the quadric 1 and P P, where
P is the projective line over F.
(b) If A is a non-singular matrix, show that
A:
42 X 3
9
1
: Ker X ) A
Im X which corresponds under this bijection to the set p p A : p
(c) Show that, if π is any permutation of P, then p π p 5 : p
set; and all special sets have this form.
9
9
P .
P is a special
(d) Deduce that every special set is a quadric if and only if F \$ 3.
8.4. Dualities of symplectic quadrangles
123
3. Consider the case n 3. Take 1 to be the Klein quadric. Show that the
Klein correspondence maps the special set to a set S of lines of PG 3 F with
@
the property that the set of lines of S through any point of p, or the set of lines
of S in any plane Π, is a plane pencil. Show that the correspondence p BD Π of
PG 3 F , where the set of lines of S containing p and the set contained in Π are
equal, is a symlectic polarity with S as its set of absolute lines. Deduce that S is
the set of lines of a symplectic GQ in PG 3 F , and hence that is a quadric.
@
4. Prove by induction on n that, for n E 3, any special set is a quadric. (See
Cameron and Kantor  for a crib.)
8.4
Dualities of symplectic quadrangles
A field of characteristic 2 is said to be perfect if every element is a square.
A finite field of characteristic 2 is perfect, since the multiplicative group has odd
order.
If F has characteristic 2, then the map x BD x2 is a homomorphism, since
x y
xy 2
2
x2 y2 x2 y2 and is one-to-one. Hence F is perfect if and only if this map is an automorphism.
Theorem 8.5 Let F be a perfect field of characteristic 2. Then there is an isomorphism between the symplectic polar space of rank n over F, and the orthogonal
polar space of rank n defined by a quadratic form in 2n 1 variables.
Proof Let V be a vector space of rank 2n 1 carrying a non-singular quadratic
form f of rank n. By polarising f , we get an alternating bilinear form b, which
cannot be non-degenerate; its radical R V : is of rank 1, and the restriction of f
to it is the germ of f .
Let W0 be a totally singular subspace of V . Then W W0 R is a totally
isotropic subspace of the non-degenerate symplectic space V R. So we have an
incidence-preserving injection θ : W0 BD
W0 R R from the orthogonal polar
space to the symplectic. We have to show that θ is onto.
So let W R be t.i. This means that W itself is t.i. for the form b; but R F W , so
W is not t.s. for f . However, on W , we have
f w1 w2 f αw f w1 4 f w2 α2 f w 124
8. The Klein quadric and triality
so f is semilinear on W . Thus, the kernel of f is a hyperplane W0 of W . The space
W0 is t.s., and W0 R W ; so W0 maps onto W R under θ.
Now consider the case n 2. We have an isomorphism between the symplectic
and orthogonal quadrangles, by Theorem 8.5, and a duality, by Theorem 8.3. So:
Theorem 8.6 The symplectic generalised quadrangle over a perfect field of characteristic 2 is self-dual.
When is there a polarity?
Theorem 8.7 Let F be a perfect field of characteristic 2. Then the symplectic GQ
over F has a polarity if and only if F has an automorphism σ satisfying
σ2
where 2 denotes the automorphism x BD
2
x2 .
Proof For this, we cannot avoid using coordinates! We take the vector space F 4
with the standard symplectic form
x1 y2 x2 y1 x3 y4 x4 y3 b x1 x2 x3 x4 y1 y2 y3 y4 (Remember that the characteristic is 2.) The Klein correspondence takes the line
spanned by x1 x2 x3 x4 and y1 y2 y3 y4 to the point with coordinates zi j , 1 i j 4, where zi j xi y j x j yi ; this point lies on the quadric with equation
z12 z34 z13 z24 z14 z23
0
and (if the line is t.i.) also on the hyperplane z12 z34 0. If we factor out
the subspace spanned by the point with z12 z34 1, zi j 0 otherwise, and use
coordinates z13 z24 z14 z23 , we obtain a point of the symplectic space; the map
δ from lines to points is the duality previously defined.
To compute the image of a point p
a1 a2 a3 a4 under the duality, take two
G t.i. lines through this point and calculate their images. If a1 and a4 are non-zero,
we can use the lines joining p to the points a1 a2 0 0 and 0 a4 a1 0 ; the im
ages are a1 a3 a2 a4 a1 a4 a2 a3 and a21 a24 0 a1a2 a3 a4 . Now the image of the
line joining these points is found to be the point a21 a22 a23 a24 . The same formula
is found in all cases. So δ2 is the collineation induced by the field automorphism
x BD x2 , or 2 as we have called it.
8.4. Dualities of symplectic quadrangles
125
Suppose that there is a field automorphism σ with σ2 2, and let θ σ 1 ;
then δθ 2 is the identity, so δθ is a polarity.
Conversely, suppose that there is a polarity. By Theorem 7.14, any collineation
g is induced by the product of a linear transformation and a uniquely defined field
automorphism θ g . Now any duality has the form δg for some collineation g;
and
θ δg 2 2θ g 2 So, if δg is a polarity, then 2θ g 2
1, whence σ
θ g 1
satisfies σ2
2.
In the case where F is a finite field GF 2m , the automorphism group of F is
cyclic of order m, generated by 2; and so there is a solution of σ2 2 if and only if
m is odd. We conclude that the symplectic quadrangle over GF 2m has a polarity
if and only if m is odd.
We now examine the set of absolute points and lines (i.e., those incident with
their image). A spread is a set S of lines such that every point lies on a unique line
of S. Dually, an ovoid in a GQ is a set O of points with the property that any line
contains a unique point of O. Note that this is quite different from the definition
of an ovoid in PG 3 F given in Section 4.4; but there is a connection, as we will
see.
Proposition 8.8 The set of absolute points of a polarity of a GQ is an ovoid, and
the set of absolute lines is a spread.
Proof Let δ be a polarity. No two absolute points are collinear. For, if x and y
are absolute points lying on the line L, then x y and Lδ would form a triangle.
Suppose that the line L contains no absolute point. Then L is not absolute, so
9
9
Lδ 8 L. Thus, there is a unique line M containing Lδ and meeting L. Then Mδ L,
so Mδ is not absolute. But L meets M, so Lδ and Mδ are collinear; hence Lδ Mδ
and L 0 M form a triangle.
The second statement is dual.
Theorem 8.9 The set of absolute points of a polarity of a symplectic GQ in
PG 3 F is an ovoid in PG 3 F .
Proof Let σ be the polarity of the GQ - , and H the polarity of the projective
space defining the GQ. By the last result, the set I of absolute points of σ is
an ovoid in - . This means that the t.i. lines are tangents to I , and the t.i. lines
through a point of I form a plane pencil. So we have to prove that any other line
of the projective space meets I in 0 or 2 points.
126
8. The Klein quadric and triality
Let X be a hyperbolic line, p a point of X 0?I , and pσ L. Then L meets the
9
9
hyperbolic line L: in a point q. Let qσ M. Since q L, we have p M; so M
9
also meets X : , in a point r. Let N rσ . Then q N, so N meets X . Also, N meets
I in a point s. The line sσ contains s and N σ
r. So s is on two lines meeting
9
X : , whence s X . So, if X 0?I,\$E 1, then X 0?I,\$E 2.
Now let p= be another point of X 0JI , and define L= and q= as before. Let K
9
be the line pq= . Then p K, so pσ L contains x K σ . Also, K meets L= , so
x is collinear with p= . But the only point of L collinear with p= is q. So x q,
independent of p= . This means that there is only one point p=K8 p in X 0<I , and
this set has cardinality 2.
Remark Over finite fields, any ovoid in a symplectic GQ is an ovoid in the
ambient projective 3-space. This is false for infinite fields. (See Exercises 2 and
3.)
Hence, if F is a perfect field of characteristic 2 in which σ2 2 for some
automorphism σ, then PG 3 F possesses symplectic ovoids and spreads. These
give rise to inversive planes and to translation planes, as described in Sections 4.1
and 4.4. For finite fields F, these are the only known ovoids other than elliptic
quadrics.
Exercises
1. Suppose that the points and lines of a GQ are all the points and some of the
lines of PG 3 F . Prove that the lines through any point form a plane pencil, and
deduce that the GQ is symplectic.
2. Prove that an ovoid I in a symplectic GQ over the finite field GF q is
an ovoid in PG 3 q . [Hint: as in Theorem 8.3.5, it suffices to prove that any
hyperbolic line meets I in 0 or 2 points. Now, if X is a hyperbolic line with
/ then X : 0LI
/ so at most half of the q2 q2 1 hyperbolic lines
0,
X 0LIG8 0,
1 2
meet I . Take any N 2 q q2 1 hyperbolic lines including all those meeting I ,
and let ni of the chosen lines meet I in i points. Prove that ∑ ni N, ∑ ini 2N,
∑ i i 1 ni 2N.]
3. Prove that, for any infinite field F, there is an ovoid of the symplectic
quadrangle over F which is not an ovoid of the embedding projective space.
8.5. Reguli and spreads
8.5
127
Reguli and spreads
We met in Section 4.1 the concepts of a regulus in PG 3 F (the set of common
transversals to three pairwise skew lines), a spread (a set of pairwise skew lines
covering all the points), a bispread (a spread containing a line of each plane), and
a regular spread (a spread containing the regulus through any three of its lines).
We now translate these concepts to the Klein quadric.
Theorem 8.10 Under the Klein correspondence,
(a) a regulus corresponds to a conic, the intersection if 1 with a non-singular
plane Π, and the opposite regulus to the intersection of 1 with Π : ;
(b) a bispread corresponds to an ovoid, a set of pairwise non-perpendicular
points meeting every plane on 1 ;
(c) a regular spread corresponds to the ovoid 1L0 W : , where W is a line disjoint
from 1 .
Proof (a) Take three pairwise skew lines. They translate into three pairwise nonperpendicular points of 1 , which span a non-singular plane Π (so that 1M0 Π is
a conic C). Now Π : is also a non-singular plane, and 1N0 Π : is a conic C = , consisting of all points perpendicular to the three given points. Translating back, C =
corresponds to the set of common transversals to the three given lines. This set is
a regulus, and is opposite to the regulus spanned by the given lines (corresponding
to C).
(b) This is straightforward translation. Note, incidentally, that a spread (or a
cospread) corresponds to what might be called a “semi-ovoid”, were it not that this
term is used for a different concept: that is, a set of pairwise non-perpendicular
points meeting every plane in one family on 1 .
(c) A regular spread is “generated” by any four lines not contained in a regulus,
in the sense that it is obtained by repeatedly adjoining all the lines in a regulus
through three of its lines. On 1 , the four given lines translate into four points, and
the operation of generation leaves us within the 3-space they span. This 3-space
has the form W : for some line W ; and no point of 1 can be perpendicular to
every point of such a 3-space.
Note that a line disjoint from 1 is anisotropic; such lines exist if and only if
there is an irreducible quadratic over F, that is, F is not quadratically closed. (We
128
8. The Klein quadric and triality
saw earlier the construction of regular spreads: if K is a quadratic extension of F,
take the rank 1 subspaces of a rank 2 vector space over K, and restrict scalars to
F.)
Thus a bispread is regular if and only if the corresponding ovoid is contained
in a 3-space section of 1 . A bispread whose ovoid lies in a 4-space section of
1 is called symplectic, since its lines are totally isotropic with respect to some
symplectic form (by the results of Section 8.3). An open problem is to find a
simple structural test for symplectic bispreads (resembling the characterisation of
regular spreads in terms of reguli).
We also saw in Section 4.1 that spreads of lines in projective space give rise
to translation planes; and regular spreads give Desarguesian (or Pappian) planes.
Another open problem is to characterise the translation planes arising from symplectic spreads or bispreads.
8.6
Triality
Now we increase the rank by 1, and let 1 be a hyperbolic quadric in PG 7 F ,
defined by a quadratic form of rank 4. The maximal t.s. subspaces have dimension
3, and are called solids; as usual, they fall into two families O 1 and O 2 , so that
two solids in the same family meet in a line or are disjoint, while two solids in
different families meet in a plane or a point. Any t.s. plane lies in a unique solid
of each type. Let P and Q be the sets of points and lines.
Consider the geometry defined as follows.
.
.
The POINTs are the elements of O
1.
.
The LINEs are the elements of Q .
.
The PLANEs are incident pairs p M , p
The SOLIDs are the elements of PSRTO
9
P ,M
9
O
2.
2.
Incidence is defined as follows. Between POINTs, LINEs and SOLIDs, it is as
in the quadric, with the additional rule that the POINT M1 and SOLID M2 are
incident if they intersect in a plane. The PLANE p M is incident with all those
varieties incident with both p and M.
Proposition 8.11 The geometry just described is an abstract polar space in which
any PLANE is incident with just two SOLIDs.
8.7. An example
129
Proof We consider the axioms in turn.
9
(P1): Consider, for example, the SOLID M O 2 . The POINTs incident with
M are bijective with the planes of M; the LINEs are the lines of M; the PLANEs
9
are pairs p M with p M, and so are bijective with the points of M. Incidence is
defined so as to make the subspaces contained in M a projective space isomorphic
to the dual of M.
9
For the SOLID p P , the argument is a little more delicate. The geometry
p :U p is a hyperbolic quadric in PG 5 F , that is, the Klein quadric; the POINTs,
LINEs and PLANEs incident with p are bijective with one family of planes, the
lines, and the other family of planes on the quadric; and hence (by the Klein
correspondence) with the points, lines and planes of PG 3 F .
The other cases are easier.
9
(P2) is trivial, (P3) routine, and (P4) is proved by observing that if p P and
9
M O 2 are not incident, then no POINT can be incident with both.
Finally, the SOLIDs containing the PLANE p M are p and M only.
So the new geometry we constructed is itself a hyperbolic quadric in PG 7 F ,
and hence isomorphic to the original one. This implies the existence of a map τ
which carries Q to itself and PVDWO 1 DWO 2 DXP . This map is called a triality of
the quadric, by analogy with dualities of projective spaces.
It is more difficult to describe trialities in coordinates. An algebraic approach
must wait until Chapter 10.
Exercise
1. Prove the Buekenhout-Shult property for the geometry constructed in this
9
9
section. That is, let M O 1 , L Q , and suppose that L is not incident with M;
prove that either all members of O 1 containing L meet M in a plane, or just one
does, depending on whether L is disjoint from M or not.
8.7
An example
In this section we apply triality to the solution of a combinatorial problem first
posed and settled by Breach and Street . Our approach follows Cameron and
Praeger .
Consider the set of planes of AG 3 2 . They form a 3- 8 4 1 design, that is,
a collection of fourteen 4-subsets of an 8-set, any three points contained in exactly
70 4-subsets altogether; can they be partitioned into
one of them. There are Y 84Z
130
8. The Klein quadric and triality
five copies of AG 3 2 ? The answer is “no”, as has been known since the time of
Cayley. (In fact, there cannot be more than two disjoint copies of AG 3 2 on an
8-set; a construction will be given in the next chapter.) Breach and Street asked:
what if we take a 9-set? This has Y 94 Z
126 4-subsets, and can conceivably be
partitioned into nine copies of AG 3 2 , each omitting one point. They proved:
Theorem 8.12 There are exactly two non-isomorphic ways to partition the 4subsets of a 9-set into nine copies of AG 3 2 . Both admit 2-transitive groups.
Proof First we construct the two examples.
1. Regard the 9-set as the projective line over GF 8 . If any point is designated
as the point at infinity, the remaining points form an affine line over GF 8 , and
hence (by restricting scalars) an affine 3-space over GF 2 . We take the fourteen
planes of this affine 3-space as one of our designs, and perform the same construction for each point to obtain the desired partition. This partition is invariant
under the group PΓL 2 8 , of order 9 8 7 3 1512. The automorphism group
is the stabiliser of the object in the symmetric group; so the number of partitions
isomorphic to this one is the index of this group in S9 , which is 9! 1512 240.
2. Alternatively, the nine points carry the structure of affine plane over GF 3 .
Identifying one point as the origin, the structure is a rank 2 vector space over
GF 3 . Put a symplectic form b on the vector space. Now there are six 4-sets
which are symmetric differences of two lines through the origin, and eight 4-sets
of the form v [R? w : b v w 1 for non-zero v. It is readily checked that
these fourteen sets form a 3-design. Perform this construction with each point
designated as the origin to obtain a partition. This one is invariant under the
group ASL 2 3 generated by the translations and Sp 2 3 SL 2 3 , of order
9 8 3 216, and there are 9! 216 1680 partitions isomorphic to this one.
Now we show that there are no others. We use the terminology of coding theory. Note that the fourteen words of weight 4 supporting planes of AG 3 2 , to
gether with the all-0 and all-1 words, form the extended Hamming code of length
8 (the code we met in Section 3.2, extended by an overall parity check); it is the
only doubly-even self-dual code of length 8 (that is, the only code C C : with
all weights divisible by 4).
Let V be the vector space of all words of length 9 and even weight. The
1
function f v mod 2 is a quadratic form on V , which polarises to
wt v 2 the usual dot product. Thus maximal t.s. subspaces for f are just doubly even
self-dual codes, and their existence shows that f has rank 4 and so is the split
8.8. Generalised polygons
131
form defining the triality quadric. (The quadric 1 consists of the words of weight
4 and 8.)
Suppose we have a partition of the 4-sets into nine affine spaces. An easy
counting argument shows that every point is excluded by just one of the designs.
So if we associate with each design the word of weight 8 whose support is its
point set, we obtain a solid on the quadric, and indeed a spread or partition of the
quadric into solids.
All these solids belong to the same family, since they are pairwise disjoint. So
we can apply the triality map and obtain a set of nine points which are pairwise
non-collinear, that is, an ovoid. Conversely, any ovoid gives a spread. In fact, an
ovoid gives a spread of solids of each family, by applying triality and its inverse.
So the total number of spreads is twice the number of ovoids.
The nine words of weight 8 form an ovoid. Any ovoid is equivalent to this
one. (Consider the Gram matrix of inner products of the vectors of an ovoid; this
must have zeros on the diagonal and ones elsewhere.) The stabiliser of this ovoid
is the symmetric group S9 . So the number of ovoids is the index of S9 in the
orthogonal group, which turns out to be 960. Thus, the total number of spreads is
1920 240 1680, and we have them all!
8.8
Generalised polygons
Projective and polar spaces are important members of a larger class of geometries called buildings. Much of the importance of these derives from the fact that
they are the “natural” geometries for arbitrary groups of Lie type, just as projective
spaces are for linear groups and polar spaces for classical groups. The groups of
Lie type include, in particular, all the non-abelian finite simple groups except for
the alternating groups and the twenty-six sporadic groups. I do not intend to discuss buildings here — for this, see the lecture notes of Tits [S] or the recent books
by Brown [C] and Ronan [P] — but will consider the rank 2 buildings, or generalised polygons as they are commonly known. These include the 2-dimensional
projective and polar spaces (that is, projective planes and generalised quadrangles).
Recall that a rank 2 geometry has two types of varieties, with a symmetric
incidence relation; it can be thought of as a bipartite graph. We use graph-theoretic
terminology in the following definition. A rank 2 geometry is a generalised n-gon
(where n E 2) if
(GP1) it is connected with diameter n and girth 2n;
132
8. The Klein quadric and triality
(GP2) for any variety x, there is a variety y at distance n from x.
It is left to the reader to check that, for n 2 3 4, this definition coincides
with that of a digon, generalised projective plane or generalised quadrangle respectively.
Let - be a generalised n-gon. The flag geometry of - has as POINTs the
varieties of - (of both types), and as LINEs the flags of - , with the obvious
incidence between POINTs and LINEs. It is easily checked to be a generalised
2n-gon in which every line has two points; and any generalised 2n-gon with two
points per line is the flag geometry of a generalised n-gon. In future, we usually
assume that our polygons are thick, that is, have at least three varieties of one
type incident with each variety of the other type. It is also easy to show that a
thick generalised polygon has orders, that is, the number of points per line and
the number of lines per point are both constant; and, if n is odd, then these two
constants are equal. [Hint: in general, if varieties x and y have distance n, then
each variety incident with x has distance n 2 from a unique variety incident with
y, and vice versa.]
We let s 1 and t 1 denote the numbers of points per line or lines per point,
respectively, with the proviso that either or both may be infinite. (If both are finite,
then the geometry is finite.) The geometry is thick if and only if s t \ 1. The major
theorem about finite generalised polygons is the Feit–Higman theorem (Feit and
Higman :
Theorem 8.13 A thick generalised n-gon can exist only for n
2 3 4 6 or 8.
In the course of the proof, Feit and Higman derive additional information:
.
if n
.
if n
6, then st is a square;
8, then 2st is a square.
Subsequently, further numerical restrictions have been discovered; for example:
.
if n
.
if n
4 or n
8, then t s2 and s t 2 ;
6, then t s3 and s t 3 .
In contrast to the situation for n 3 and n 4, the only known finite thick
generalised 6-gons and 8-gons arise from groups of Lie type. There are 6-gons
8.9. Some generalised hexagons
133
with s t q and with s q, t q3 for any prime power q; and 8-gons with
s q, t q2 , where q is an odd power of 2. In the next section, we discuss a class
of 6-gons including the first-mentioned finite examples.
There is no hope of classifying infinite generalised n-gons, which exist for all
n (Exercise 2). However, assuming a symmetry condition, the Moufang condition,
which generalises the existence of central collineations in projective planes, and
is also equivalent to a generalisation of Desargues’ theorem, Tits [35, 36] and
Weiss  derived the same conclusion as Feit and Higman, namely, that n 2,
3, 4, 6 or 8.
As for quadrangles, the question of the existence of thick generalised n-gons
(for n E 3) with s finite and t infinite is completely open. Of course, n must be
even in such a geometry!
Exercises
1. Prove the assertions claimed to be “easy” in the text.
2. Construct infinite “free” generalised n-gons for any n E 3.
8.9
Some generalised hexagons
In this section, we use triality to construct a generalised hexagon called G2 F over any field F. The construction is due to Tits. The name arises from the fact
that the automorphism groups of these hexagons are the Chevalley groups of type
G2 , as constructed by Chevalley from the simple Lie algebra G2 over the complex
numbers.
We begin with the triality quadric 1 . Let v be a non-singular vector. Then
v : 0?1 is a rank 3 quadric. Its maximal t.s. subspaces are planes, and each lies in
a unique solid of each family on 1 . Conversely, a solid on 1 meets v : in a plane.
Thus, fixing v, there is are bijections between the two families of solids and the
set of planes on 1 =
1A0 v : . On this set, we have the structure of the dual polar
space induced by the quadric 1 = ; in other words, the POINTs are the planes on
this quadric, the LINES are the lines, and incidence is reversed inclusion. Call
this geometry - .
Applying triality, we obtain a representation of - using all the points and
some of the lines of 1 .
Now we take a non-singular vector, which may as well be the same as the
vector v already used. (Since we have applied triality, there is no connection.)
134
8. The Klein quadric and triality
The geometry ] consists of those points and lines of - which lie in v : . Thus, it
consists of all the points, and some of the lines, of the quadric 1^= .
Theorem 8.14 ]
is a generalised hexagon.
Proof First we observe some properties of the geometry - , whose points and
lines correspond to planes and lines on the quadric 1_= . The distance between two
points is equal to the codimension of their intersection. If two planes of 1 = meet
non-trivially, then the corresponding solids of 1 (in the same family) meet in a
line, and so (applying triality) the points are perpendicular. Hence:
(a) Points of - lie at distance 1 or 2 if and only if they are perpendicular.
Let x y z w be four points of - forming a 4-cycle. These points are pairwise
perpendicular (by (a)), and so they span a t.s. solid S. We prove:
(b) The geometry induced on S by - is a symplectic GQ.
Keep in mind the following transformations:
solid S
D point p (by triality)
D quadric 1 ¯ in p : p (residue of p)
D PG 3 F (Klein correspondence).
Now points of S become solids of one family containing p, then planes of one
family in 1 ¯ , then points in PG 3 F ; so we can identify the two ends of this chain.
Lines of - in S become lines through p perpendicular to v, then points of 1 ¯
2 v p 3 p, then t.i. lines of a symplectic GQ, by the correperpendicular to 2 v̄ 3
spondence described in Section 8.3. Thus (b) is proved.
A property of - established in Proposition 7.9 is:
(c) If x is a point and L a line, then there is a unique point of L nearest to x.
We now turn our attention to ]
(d) Distances in ]
, and observe first:
are the same as in - .
For clearly distances in ] are at least as great as those in - , and two points of ]
at distance 1 (i.e., collinear) in - are collinear in ] .
9
Suppose that x y ] lie at distance 2 in - . They are joined by more than
one path of length 2 there, hence lie in a solid S carrying a symplectic GQ, as
8.9. Some generalised hexagons
135
in (b). The points of ] in S are those of S 0 v : , a plane on which the induced
substructure is a plane pencil of lines of ] . Hence x and y lie at distance 2 in ] .
9
Finally, let x y ] lie at distance 3 in - . Take a line L of ] through y; there
is a point z of - (and hence of ] ) on L at distance 2 from x (by (c)). So x and y
lie at distance 3 in ] .
In particular, property (c) holds also in ] .
(e) For any point x of ]
, the lines of ]
through x form a plane pencil.
For, by (a), the union of these lines lies in a t.s. subspace, hence they are coplanar;
there are no triangles (by (c)), so this plane contains two points at distance 2; now
the argument for (d) applies.
Finally:
(f) ]
is a generalised hexagon.
We know it has diameter 3, and (GP2) is clearly true. A circuit of length less than
6 would be contained in a t.s. subspace, leading to a contradiction as in (d) and
(e). (In fact, by (c), it is enough to exclude quadrangles.)
Cameron and Kantor  give a more elementary construction of this hexagon.
Their construction, while producing the embedding in 1^= , depends only on properties of the group PSL 3 F . However, the proof that it works uses both counting
arguments and arguments about finite groups; it is not obvious that it works in
general, although the result remains true.
If F is a perfect field of characteristic 2 then, by Theorem 8.5, 1`= is isomorphic
to the symplectic polar space of rank 3; so ] is embedded as all the points and
some of the lines of PG 5 F .
Two further results will be mentioned without proof. First, if the field F has an
automorphism of order 3, then the construction of ] can be “twisted”, much as
can be done to the Klein correspondence to obtain the duality between orthogonal
and unitary quadrangles (mentioned in Section 8.3), to produce another generalised hexagon, called 3D4 F . In the finite case, 3D4 q3 has parameters s q3 ,
t q.
Second, there is a construction similar to that of Section 8.4. The generalised
hexagon G2 F is self-dual if F is a perfect field of characteristic 3, and is self
polar if F has an automorphism σ satisfying σ2 3. In this case, the set of absolute
points of the polarity is an ovoid, a set of pairwise non-collinear points meeting
every line of ] , and the group of collineations commuting with the polarity has
as a normal subgroup the Ree group 2G2 F , acting 2-transitively on the points of
the ovoid.
136
8. The Klein quadric and triality
Exercise
1. Show that the hexagon ] has two disjoint planes E and F, each of which
consists of pairwise non-collinear (but perpendicular) points. Show that each point
of E is collinear (in ] ) to the points of a line of F, and dually, so that E and F
are naturally dual. Show that the points of E R F, and the lines of ] joining
their points, form a non-thick generalised hexagon which is the flag geometry of
PG 2 F . (This is the starting point in the construction of Cameron and Kantor
referred to in the text.)
9
The geometry of the Mathieu groups
The topic of this chapter is something of a diversion, but is included for two reasons: first, its intrinsic interest; and second, because the geometries described here
satisfy axioms not too different from those we have seen for projective, affine and
polar spaces, and so they indicate the natural boundaries of the theory.
9.1
The Golay code
The basic concepts of coding theory were introduced in Section 3.2, where
we also saw that a non-trivial perfect 3-error-correcting code must have length 23
(see Exercise 3.2.2). Such a code C may be assumed to contain the zero word (by
translation), and so any other word has weight at least 7; and
C
23
0 223
23
1 23
2 23
3
212 We extend C to a code C of length 24 by adding an overall parity check; that
is, we put a 0 in the 24th coordinate of a word whose weight (in C) is even, and a 1
in a word whose weight is odd. The resulting code has all words of even weight,
and hence all distances between words even; since adding a coordinate cannot
decrease the distance between words, the resulting code has minimum distance 8.
In this section, we outline a proof of the following result.
Theorem 9.1 There is a unique code with length 24, minimum distance 8, and
containing 212 codewords one of which is zero (up to coordinate permutations).
This code is known as the (extended binary) Golay code. It is a linear code
(the linearity does not have to be assumed).
137
138
9. The geometry of the Mathieu groups
Remark There are many constructions of this code; for an account of some of
these, see Cameron and Van Lint [F]. As a general principle, a good construction
of an object leads to a proof of its uniqueness (by showing that it must be constructed this way), thence to a calculation of its automorphism group (since the
object is uniquely built around a starting configuration, and so any isomorphism
between such starting configurations extends uniquely to an automorphism), and
gives on the way a subgroup of the automorphism group (consisting of the automorphism group of the starting configuration). This point will not be laboured
below, but the interested reader may like to examine this and other constructions
from this point of view. The particular construction given here has been chosen
for two reasons: first, as an application of the Klein correspondence; and second,
since it makes certain properties of the automorphism group more accessible.
Proof First, we review the isomorphism between PSL 4 2 and A8 outlined in
Exercise 8.1.1. Let U be the binary vector space consisting of words of even
weight and length 8, Z the subspace consisting of the all-zero and all-one words,
and V U Z. The function mapping a word of U to 0 or 1 according as its
weight is congruent to 0 or 2 mod 4 induces a quadratic form f on V , whose zeros
form the Klein quadric ; let W be the vector space of rank 4 whose lines are
bijective
with the points of . Note that the points of correspond to partitions
of N
1 8 into two subsets of size 4.
Let Ω N W . This set will index the coordinates of the code C we construct.
A words of C will be specified by its support, a subset of N and a subset of W . In
particular, 0/ N W and N W will be words; so we can complement the subset of
N or the subset of W defining a word and obtain another word.
The first non-trivial class of words is obtained by combining the empty subset
of N (or the whole of N) with any hyperplane in W (or its coset).
A complementary pair of 4-subsets of N corresponds to a point of , and
hence to a line L in W . Each 4-subset of N, together with any coset of the corresponding L, is a codeword. Further words are obtained by replacing the coset of
L by its symmetric difference with a coset of a hyperplane not containing L (such
a coset meets L in two vectors).
A 2-subset of N, or the complementary 6-subset, represents a non-singular
point, which translates into a symplectic form b on W . The quadric associated
with any quadratic form which polarises to b, together with the 2-subset of N,
defines a codeword.
9.2. The Witt system
139
This gives us a total of
4 4 15 8
4 4 7 4
8
12
16 4 2
2
codewords. Moreover, a fairly small amount of case checking shows that the code
is linear. Its minimum weight is visibly 8.
We now outline the proof that there is a unique code C of length 24, cardinality
212 , and minimum weight 8, containing 0. Counting arguments show that such a
code contains 759 words of weight 8, 2576 of weight 12, 759 of weight 16, and
the all-1 word 1 of weight 24. Now, if the code is translated by any codeword,
the hypotheses still hold, and so the conclusion about weights does too. Thus,
the distances between pairs of codewords are 0, 8, 12, 16, and 24. It follows that
all inner products are zero, so C C ; it then follows from the cardinality that
C C , and in particular C is a linear code.
Let N be an octad, and W its complement. Restriction of codewords to N
gives a homomorphism θ from C to a code of length 8 in which all words have
even weight. It is readily checked that every word of even weight actually occurs.
So the kernel of θ has rank 5. This kernel is a code of length 16 and minimum
weight 8. There is a unique code with these properties: it consists of the all-zero
and all-one words, together with the characteristic functions of hyperplanes of a
rank 4 vector space. (This is the first-order Reed–Muller code of length 16.) Thus
we have identified W with a vector space, and found the first non-trivial class of
words in the earlier construction.
Now, to be brief: if B is an octad meeting N in four points, then B W is a line;
if B N
2, then B W is a quadric; and all the other details can be checked,
given sufficient perseverence.
The automorphism group of the extended Golay code is the 54-transitive Mathieu group M24 . This is one of only two finite 5-transitive groups other than symmetric and alternating groups; it is one of the first of the 26 “sporadic” simple
groups to be found; and its geometry is the starting point for the construction of
many other sporadic groups (the Conway and Fischer groups and the “Monster”).
The group M24 will be considered further in Section 9.4.
9.2
The Witt system
Let X be the set of coordinate positions of the Golay code G. Now any word
can be identified uniquely with the subset of X consisting of the positions where
140
9. The geometry of the Mathieu groups
it has entries equal to 1 (its support). Let be the set of supports of the 759
codewords of weight 8. An element of is called an octad; the support of a word
of weight 12 in G is called a dodecad.
From the linearity of G, we see that the symmetric difference of two octads is
the support of a word of G, necessarily an octad, a dodecad, or the complement of
an octad; the intersection of the two octads has cardinality 4, 2 or 0 respectively.
Three pairwise disjoint octads form a trio. (In our construction of the extended
Golay code in the last section, the three “blocks” of eight coordinates form a trio.)
Proposition 9.2 X is a 5- 24 8 1 design or Steiner system.
Proof As we have just seen, it is impossible for two octads to have more than
four points in common, so five points lie in at most one octad.
are 759
8 Since
24 there
octads, the average number containing five points is 759 5 5 1; so five
points lie in exactly one octad. However, the proposition follows more directly
from the properties of the code G.
Take any five coordinates, and delete one of them. The remaining coordinates
support a word v of weight 4. But the Golay code obtained by deleting a coordinate from G is perfect 3-error-correcting, and so contains a unique word c at
distance 3 or less from v. It must hold that c has weight 7 and its support contains
that of v (and c is the unique such word). Re-introducing the deleted coordinate
(which acts as a parity check for the Golay code), we obtain a unique octad containing the given 5-set.
This design is known as the Witt system; Witt constructed it from its automorphism group, the Mathieu group M24 , though nowdays the procedure is normally
reversed.
Now
choose any three coordinates, and call them ∞1 , ∞2 , ∞3 . Let X X ∞1 ∞2 ∞3 , and let be the set of octads containing the chosen points,
with these points removed. Then X is a 2-(21, 5, 1) design, that is, a projective plane of order 4. Since there is a unique projective plane of order 4 (see
Exercise 4.3.6), it is isomorphic to PG 2 4 .
Proposition 9.3 The geometry whose varieties are all subsets of X of cardinalities
1, 2, 3 and 4, and all octads, with incidence defined by inclusion, belongs to the
diagram
c
c
c
9.2. The Witt system
141
The remaining octads can be identified with geometric configurations in PG 2 4 .
We outline this, omitting detailed verification. In fact, the procedure can be reversed, and the Witt system constructed from objects in PG 2 4 . See Lüneburg [N]
for the details of this construction.
1. An octad containing two of the three points ∞i corresponds to a set of six
points of PG 2 4 meeting any line in 0 or 2 points, in other words, a hyperoval.
All 168 hyperovals occur in this way. If we call two hyperovals “equivalent” if
their intersection has even cardinality, we obtain a partition into three classes of
size 56, corresponding to the three possible pairs of points ∞i ; so this partition can
be defined internally.
2. An octad containing one point ∞i corresponds to a set of seven points
of PG 2 4 meeting every line in 1 or 3 points, that is, a Baer subplane (when
equipped with the lines meeting it in three points). Again, all 360 Baer subplanes
occur, and the partition can be intrinsically defined.
3. An octad containing none of the points ∞i is a set of eight points of PG 2 4 which is the symmetric difference of two lines. Every symmetric difference of two
lines occurs (there are 210 such sets).
Since octads and dodecads also intersect evenly, we can extend this analysis
to dodecads. Consider a dodecad containing ∞1 , ∞2 and ∞3 . It contains nine
points of PG 2 4 , meeting every line in 1 or 3 points. These nine points form
a unital, the set of absolute points of a unitary polarity (or the set of zeros of a
non-degenerate Hermitian form). Their intersections of size 3 with lines form a
2- 9 3 1 design, a Steiner triple system which is isomorphic to AG 2 3 , and
is also famous as the Hessian configuration of inflection points of a non-singular
cubic. (Since the field automorphism of GF 4 is α ! α2 , the Hermitian form
x0 xα1 x1 xα0 x2 xα2 is a cubic form, and its zeros form a cubic curve; in this special
case, every point is an inflection.)
Exercises
1. Verify the connections between octads and dodecads and configurations in
PG 2 4 claimed in the text.
2. Let B be an octad, and Y X B. Consider the geometry " whose points
are those of Y ; whose lines are all pairs of points; whose planes are all sets B B,
where B is an octad meeting B in four points; and whose solids are the octads
disjoint from B. prove that " is the affine geometry AG 4 2 .
142
9.3
9. The geometry of the Mathieu groups
Sextets
A tetrad is a set of four points of the Witt system. Any tetrad is contained
in five octads, which partition the remaining twenty points into five tetrads. Now
the symmetric difference of two octads intersecting in a tetrad is an octad; so the
union of any two of our six tetrads is an octad. A set of six pairwise disjoint tetrads
with this property is called a idxsextet.
Proposition 9.4 Let " be the geometry whose POINTS, LINES and PLANES are
the octads, trios and sextets respectively, with incidence defined as follows: a
LINE is incident with any POINT it contains; a PLANE is incident with a POINT
which is the union of two of its tetrads; and a PLANE is incident with a LINE if it
is incident with each POINT of the LINE. Then " belongs to the diagram
where
L
L
is the linear space consisting of points and lines of PG 3 2 .
Proof Calculate residues. Take first a PLANE or sextet. It contains six tetrads;
the union of any two of them is a POINT, and any partition into three sets of two
is a LINE. This is a representation of the unique GQ with s t 2 that we saw in
Section 7.1.
Now consider the residue of a POINT or octad. We saw in Exercise 9.2.2
that the complement of an octad carries an affine space AG 4 2 ; LINEs incident
with the POINT correspond to parallel classes of planes in the affine space, and
PLANEs incident with it to parallel classes of LINEs. Projectivising and dualising, we see the points and lines of PG 3 2 .
Finally, any POINT and PLANE incident with a common LINE are incident
with one another.
The geometry does not contain objects which would correspond to the planes
of PG 3 2 in the residue of a point. The diagram is sometimes drawn with a
“ghost node” corresponding to these non-existent varieties.
Exercise
1. In the geometry " of Proposition 9.4, define the distance between two
points to be the number of lines on a shortest path joining them. Prove that, if x is
a point and L a line, then there is a unique point of L at minimum distance from x.
9.4. The large Mathieu groups
9.4
143
The large Mathieu groups
Just as every good construction of the Golay code or the Witt system contains
the seeds of a uniqueness proof (as we observed in Section 9.1), so every good
uniqueness proof contains the seeds of an argument establishing various properties
of its automorphism group (in particular, its order, and some large subgroup, the
particular subgroup depending on the construction used). I will outline this for the
construction of Section 9.1.
Theorem 9.5 The automorphism group of the Golay code, or of the Witt system,
is a 5-transitive simple group of order 24 23 22 21 20 48 Remark This group is of course the Mathieu group M24 . Part of the reason for
the construction we gave (not the simplest available!) is that it makes our job now
easier.
Proof First note that the design and the code have the same automorphism group;
for the code is spanned by the design, and the design is the set of words of weight
8 in the code.
The uniqueness proof shows that the automorphism group is transitive on octads. For, given two copies of the Golay code, and an octad in each, there is an
isomorphism between the two codes mapping the chosen octad in the first to that
in the second. Also, the stabiliser of an octad preserves the affine space structure
on its complement, and (from the construction) induces AGL 4 2 on it. (It induces A8 on the octad, the kernel of this action being the translation group of the
affine space.) This gives the order of the group.
Given two 5-tuples of distinct points, each lies in a unique octad. There is an
automorphism carrying the first octad to the second; then, since A8 is 5-transitive,
we can fix the second octad and map the 5-tuple to the correct place. The 5transitivity follows.
We also have a subgroup H AGL 4 2 of our unknown group G, and it is
easily seen that
H is maximal. Suppose that N is a non-trivial normal subgroup of
G. Then HN G, and H N is a normal subgroup
of H, necessarily the identity or
the translation group. (If H N H then N G.) This gives two possibilities for
the order of N, namely 759 and 759 16. But N, a normal subgroup of a 5-transitive
group, is at least 4-transitive, by an old theorem of Jordan; so 24 23 22 21 divides
N , a contradiction. We conclude that G is simple.
144
9. The geometry of the Mathieu groups
The stabiliser of three points is a group of collineations of PG 2 4 , necessarily PSL 3 4 (by considering order). The ovals and Baer subplanes each fall
into three orbits for PSL 3 4 , these orbits being the classes used in Lüneburg’s
construction. The set-wise stabiliser of three points is PΓL 3 4 . Looked at another way, Lüneburg’s construction and uniqueness proof gives us the subgroup
PΓL 3 4 of M24 .
9.5
The small Mathieu groups
To conclude this chapter, I describe briefly the geometry associated with the
Mathieu group M12 .
There are two quite different approaches. One locates the geometry within the
Golay code. The group M12 can be defined as the stabiliser of a dodecad in M24 ;
it acts sharply 5-transitively on this dodecad, and on the complementary dodecad,
but the two permutation representations are not equivalent. The dodecad D carries
a design, which can be seen as follows. It intersects any octad in an even number,
at most 6, of points; and any five points of D lie in a unique octad, meeting D
in 6 points. So the intersections of size 6 of octads with D are the blocks of a
5- 12 6 1 design or Steiner system.
Alternatively, there are “characteristic 3” objects with properties resembling
the binary Golay code. There is a ternary Golay code, a set of ternary words of
length 12 (that is, entries in GF 3 ) forming a subspace of GF 3 12 of rank 6, and
having minimum weight 6; the supports of weight 6 of codewords form the blocks
of the design. Alternatively, there is a set of 12 points in PG 5 3 on which M12 is
induced, as follows. There is a Hadamard matrix H of size 12 # 12 (a matrix with
entries \$ 1 satisfying HH %
12I), unique up to row and column permutations
and sign changes; over GF 3 , it has rank 6, and its rows span the required points.
Now the design is obtained as follows. The point set is identified with the set of
rows. Any two columns agree in six rows and disagree in the
six, defining
12 other
132 blocks are
two sets of size 6 which are blocks of the design; and all 2 2 obtained in this way.
Some connection between characteristics 2 and 3 can be seen from the observation we made in Section 9.2, that a unital in PG 2 4 is isomorphic to the affine
plane AG 2 3 . It turns out that the three times extensions of these two planes
are associated with codes in characteristics 2 and 3 respectively, and that one extension contains the other. However, the large Witt system is not embeddable in
PG 5 4 , so the analogy is not perfect.
9.5. The small Mathieu groups
145
Exercise
1. Let G AG 2 3 , and X the set of lines of G (so that X
the subsets of X of the following types:
'
&
12). Consider
'
all unions of two parallel classes;
'
the lines of two classes containing a point p, and those of the other two not
containing p;
a parallel class, with the lines of the others containing a fixed point p; and
the complements of these.
Show that these 6 54 2 36 132 sets of size 6 form a 5- 12 6 1 design.
Assuming the uniqueness of this design, prove that AGL 2 3 )( M12 .
10
Exterior powers and Clifford
algebras
In this chapter, various algebraic constructions (exterior products and Clifford algebras) are used to embed some geometries related to projective and polar spaces
(subspace and spinor geometries) into projective spaces. In the process, we learn
more about the geometries themselves.
10.1
Tensor and exterior products
Throughout this chapter, F is a commutative field (except for a brief discussion
of why this assumption is necessary).
The tensor product V W of two F-vector spaces V and W is the free-est
bilinear product of V and W : that is, if (as customary), we write the product of
vectors v V and w W as v w, then we have
αv w α v
v αw α v
v1 v2 w v1 w v2 w v w1 w2 v w1 v w2 w w Formally,
we let X be the F-vector space with basis consisting of all the ordered
pairs v w (v V w W ), and
Y the subspace spanned by all expressions of the
form v1 v2 w v1 w v 2 w and three similar expressions; then V W X Y , with v w the image of v w under the canonical projection. Sometimes,
to emphasize the field, we write V F W .
This construction will only work as intended over a commutative field. For
αβ v
w α βv
w βv
αw β v
147
αw βα v
w 148
10. Exterior powers and Clifford algebras
so if v w 0 then αβ βα.
There are two representations
convenient for calculation. If V has a basis
v1 vn and W a basis w1 wm , then V W has a basis
w j : 1 i n 1 vi
j m If V and W are identified with F n and F m respectively, then V W can be
identified with the space of n m matrices over F, where v w is mapped to the
matrix v w.
In particular, rk V W rk V rk W .
Suppose that V and W are F-algebras (that is, have an associative multiplication which is compatible with the vector space structure). Then V W is an
algebra, with the rule
v1
w2 w1 v2
v1 v2 w1 w2 Of course, we can form the tensor product of a space with itself; and we can
form iterated tensor products of more than two spaces. Let k V denote the k-fold
tensor power of V . Now the tensor algebra of V is defined to be
∞ T V k 0
k
V with multiplication given by the rule
v1
vn vn 1
vm n v1
vm n
on homogeneous elements, and extended linearly. It is the free-est associative
algebra generated by V .
The exterior square of a vector space V is the free-est bilinear square of V in
which the square of any element of V is zero. In other words, it is the quotient of
2 V by the subspace generated by all vectors v v for v V . We write it as 2 V ,
or
V V , and denote the product of v and w by v w. Note that w v v w. If
v1 vn is a basis for V , then a basis for V V consists of all vectors v1 v j ,
for 1 i j n; so
rk V V n
2
1 2n
n 1 More generally, we can define the kth exterior power k V as a k-fold multilinear product, in which any product of vectors vanishes if two factors are equal.
10.1. Tensor and exterior products
Its basis consists of all expressions vi1 vik , with 1 i1 its dimension is ! nk " . Note that k V 0 if k # n rk V .
The exterior algebra of V is\$
n V k 0
149
ik n; and
kV with multiplication defined as for the tensor algebra. Its rank is ∑nk 0 ! nk" 2n .
If θ is a linear transformation on V , then θ induces in a natural
way linear
transformations
k θ on k V , and k θ on k V , for all k. If rk V n, then we
have rk n V 1, and so n θ is a scalar. In fact, n θ det θ . (This fact is the
basis of an abstract, matrix-free, definition of the determinant.)
Exercises
1. Let F be a skew field, V a right F-vector space, and W a left vector space.
Show that it is possible to define V F W as an abelian group so that
v1 v2 and
w v1
w v2
vα w
v
w v
w1 w2 v
w1 v
w2
αw 2. In the identification of F n F m with the space of n m matrices, show
that the rank of a matrix is equal to the minimum r for which the corresponding
tensor can be expressed in the form ∑ri 1 vi wi . Show that, in such a minimal
expression, v1 vr are linearly independent, as are w1 wr .
3. (a) If K is an extension field of F, and n a positive integer, prove that
Mn F F
K % Mn K where Mn F is the ring of n n matrices over F.
('
& % &*)+& .
(b) Prove that &
4. Define the symmetric square S2V of a vector space V , the free-est bilinear
square of V in which v w w v. Find a basis for it, and calculate its dimension.
More generally, define the kth symmetric
power SkV , and calculate its dimension;
and define the symmetric algebra S V . If dim V n, show that the symmetric
algebra on V is isomorphic to the polynomial ring in n variables over the base
field.
5. Prove that, if θ is a linear map on V , where rk V n, then n θ det θ .
150
10.2
10. Exterior powers and Clifford algebras
The geometry of exterior powers
Let V be an F-vector space of rank n, and k a positive integer less
than n. There
are a couple of ways of defining a geometry on the set Σk Σk V of
subspaces
n
of V of rank k (equivalently, the k 1 -dimensional subspaces of PG ! k " 1 F ,
which I now describe.
The first approach produces a point-line geometry.
For each pair U1 U2 of
subspaces of V with U1 , U2 , rk U1 k 1, rk U2 k 1, a line
L U1 U2 W Σk : U1 , W , U2 Now two points lie in at most one line. For, if W1 W2 are distinct subspaces of
rank k and W1 W2 L U1 U2 , then U1 - W1 . W2 and / W1 W2 01- U2 ; so equality
must hold in both places. Note that two subspaces are collinear if and only if
their intersection has codimension 1 in each. We call this geometry a subspace
geometry.
In the case k 2, the points of the subspace geometry are the lines of PG n 1 F , and its lines are the plane pencils. In particular, for k 2, n 4, it is the
Klein quadric.
The subspace geometry has the following important property:
Proposition 10.1 If three points are pairwise collinear, then they are contained
in a projective plane. In particular, a point not on a line L is collinear with none,
one or all points of L.
Proof Clearly the second assertion follows from the first. In order to prove the
first assertion, note that there are two kinds of projective planes in the geometry,
W , U2 ,
consisting of all points W (i.e., subspaces of rank
k) satisfying U
1 ,
where either rk U1 k 1, rk U2 k 2, or rk U1 k 2, rk U2 k 1.
So let W1 W2 W3 be pairwise collinear points. If rk W1 . W2 . W3 k 1,
then the three points
are contained in a plane of the first type; so suppose not.
Then we have rk W1 . W2 . W3 k 2; and, by factoring out this intersection,
we may assume that k 2. In the projective space, W1 W2 W3 are now three
pairwise intersecting lines, and so are coplanar. Thus rk / W1 W2 W3 0 k 1, and
our three points lie in a plane of the second type.
A point-line geometry satisfying the second conclusion of Proposition 10.1 is
called a gamma space. Gamma spaces are a natural generalisation of polar spaces
10.2. The geometry of exterior powers
151
(in the Buekenhout–Shult sense), and this property has been used in several recent
characterisations (some of which are surveyed by Shult ).
The subspace geometries have natural embeddings in projective spaces given
by exterior powers, generalising
the Klein quadric. Let X 2 k V ; we consider
the projective space PG N 1 F based on X , where N ! nk " . This projective
space contains some distinguished points, those spanned by the vectors of the
form v1 vk , for v1 vk V . We call these pure products.
Theorem 10.2 (a) v1 vk 0 if and only if v1 vk are linearly dependent.
(b) The set of points of PG N 1 F spanned by non-zero pure products, together
with the lines meeting this set in more than two points, is isomorphic to the
subspace geometry Σk V .
Proof (a) If v1 vk are linearly independent, then they form part of a basis,
and their product is one of the basis vectors of X , hence non-zero. Conversely,
if these vectors are dependent, then one of them can be expressed in terms of the
others, and the product is zero (using linearity and the fact that a product with two
equal terms is zero).
(b) It follows from our remarks about determinants that, if v1 vk are replaced by another k-tuple with the same span, then v1 vk is multiplied by a
scalar factor, and the point of PG N 1 F it spans is unaltered. If W1 W2 , then
we can (as usual in linear algebra) choose a basis for V containing bases for both
W1 and W2 ; the corresponding pure products are distinct basis vectors of X , and
so span distinct points. The correspondence is one-to-one.
Suppose
that W1 and W2 are
collinear in the subspace geometry. then they have
bases v1 vk 3 1 w1 and v1 vk 3 1 w2 . Then the points spanned by the
vectors
v1 vk 3 1 αw1 βw2 form a line in PG N 1 F and represent all the points of the line in the subspace
geometry joining W1 and W2 .
Conversely, suppose that v1 vk and w1 wk are two pure products.
By factoring out the intersection of the corresponding subspaces, we may assume
that v1 wk are linearly independent. If k # 1, then no other vector in the span
of these two pure products is a pure product. If k 1, then the three points are
coplanar.
152
10. Exterior powers and Clifford algebras
The other natural geometry on the set Σk V is just the truncation of the projective geometry to ranks k 1 k and k 1; in other words, its varieties are the
subspaces of V of these three ranks, and incidence is inclusion. This geometry has
no immediate connection with exterior algebra; but it (or the more general form
based on any generalised projective geometry) has a beautiful characterisation due
to Sprague (1981).
Theorem 10.3 (a) The geometry just described has diagram
4
L5 4
L 4
where L 5 denotes the class of dual linear spaces.
(b) Conversely, any geometry with this diagram, in which chains of subspaces are
finite, consists of the varieties of ranks k 1 k and k 1 of a generalised
projective space of finite dimension, two varieties incident if one contains
the other.
projective space; and
Proof The residue of a variety of rank k 1 is the quotient
the residue of a variety of rank k 1 is the dual of PG k F . This establishes the
diagram.
I will not give the proof of Sprague’s theorem. the proof is by induction (hence
the need to assume finite rank). Sprague shows that it is possible to recognise
in the geometry objects corresponding to varieties of rank k 2, these objects
together with the left and centre nodes forming the diagram 4 L 564 L 4 again,
but with the dimension of the residue of a variety belonging to the rightmost node
reduced by 1. After finitely many steps, we reach the points, lines and planes of
the projective space, which is recognised by the Veblen–Young axioms.
Exercise
in Sec1. Show that the dual of the generalised hexagon G2 F constructed
tion 8.8 is embedded in the subspace geometry of lines of PG 6 F . [Hint: the
lines of the hexagon through a point x are all those containing x in a plane W x .]
10.3
Near polygons
In this section we consider certain special point-line geometries. These geometries will always be connected, and the distance between two points is the
10.3. Near polygons
153
smallest number of lines in a path joining them. A near polygon is a geometry
with the following property:
(NP) Given any point p and line L, there is a unique point of L nearest to p.
If a near polygon has diameter n, it is called a near 2n-gon.
We begin with some elementary properties of near polygons.
Proposition 10.4 In a near polygon,
(a) two points lie on at most one line;
(b) the shortest circuit has even length.
Proof (a) Suppose that lines L1 L2 contain points p1 p2 . Let q L1 . Then q is at
distance 1 from the two points p1 p2 of L2 , and so is at distance 0 from a unique
point of L2 ; that is, q L2 . So L1 - L2 ; and, interchanging these two lines, we
find that L1 L2 .
If a circuit has odd length 2m 1, then a point lies at distance m from two
points of the opposite line; so it lies at distance m 1 from some point of this line,
and a circuit of length 2m is formed.
Any generalised polygon is a near polygon; and any “non-degenerate” near
4-gon is a generalised quadrangle (see Exercise 1).
Some deeper structural properties are given in the next two theorems, which
were found by Shult and Yanushka .
Theorem 10.5 Suppose that x1 x2 x3 x4 is a circuit of length 4 in a near polygon,
at least one of whose sides contains more than two points. Then there is a unique
subspace containing these four points which is a generalised quadrangle.
A subspace of the type given by this theorem is called a quad.
Corollary 10.6 Suppose that a near polygon has the properties
(a) any line contains more than two points;
(b) any two points at distance 2 are contained in a circuit of length 4.
154
10. Exterior powers and Clifford algebras
Then the points, lines and quads form a geometry belonging to the diagram
4
4
L 4
We now assume that the hypotheses
of this Corollary apply. Let p be a point
and Q a quad. We say that the pair p Q is classical if
(a) there is a unique point x of Q nearest p;
(b) for y Q, d y p d x p 7 1 if and only if y is collinear with x.
(The point x is the “gateway” to Q from p.) An ovoid in a generalised quadrangle
is a set O of (pairwise non-collinear) points with the property that any further
point
of the quadrangle is collinear with a unique point of O. The point-quad pair
p Q is ovoidal if the set of points of Q nearest to p is an ovoid of Q.
Theorem 10.7 In a near polygon with at least three points on a line, any pointquad pair is either classical or ovoidal.
A proof in the finite case is outlined in Exercise 2.
We now give an example, the sextet geometry of Section 9.3 (which, as we
already know, has the correct diagram). Recall that the POINTs, LINEs, and
“QUADs” (as we will now re-name them) of the geometry are the octads, trios
and sextets of the Witt system. We check that this is a near polygon, and examine
the point-quad pairs.
Two octads intersect in 0, 2 or 4 points. If they are disjoint, they are contained
in a trio (i.e., collinear). If they intersect in four points, they define a sextet, and
so some octad is disjoint from both; so their
distance is 2. If they intersect in two
points, their distance is 3. Suppose that B1 B2 B3 is a trio and B an octad not in
this trio. Either B is disjoint from (i.e., collinear with) a unique octad in the trio,
or its intersections with them have cardinalities 4, 2, 2. In the latter case, it lies at
distance 2 from one POINT of the LINE, and distance 3 from the other two.
Now let B be a POINT (an octad), and S a QUAD (a sextet). The intersections
of B with the tetrads of S have the property that any two of them sum to 0, 2, 4
or 8; so they are all congruent mod 2. If the intersections have even parity, they
are 4 4 0 0 0 0 (the POINT lies in the QUAD) or 2 2 2 2 0 0 (B is disjoint from
a unique octad incident with S, and the pair is classical). If they have odd parity,
10.4. Dual polar spaces
155
they are 3 1 1 1 1 1; then B has distance 2 from the five octads containing
the
first tetrad, and distance 3 from the others. Note that in the GQ of order 2 2 ,
represented as the pairs from a 6-set, the five pairs containing an element of the
6-set form an ovoid. So B S is ovoidal in this case.
10.3.1
Exercises
1. (a) A near polygon with lines of size 2 is a bipartite graph.
(b) A near 4-gon, in which no point is joined to all others, is a generalised
quadrangle.
2. Let Q be a finite GQ with order s t, where s # 1.
(a) Suppose that the point set of Q is partitioned into three subsets A B C
such that for any line L, the values of 8 L . A 8 , 8 L . B 8 and 8 L . C 8 are either 1 s 0,
or 0 1 s. Prove that A is a singleton, and B the set of points collinear with A.
(b) Suppose that the point set of Q is partitioned into two subsets A and B
such that any line contains a unique point of A. Prove that A is an ovoid.
(c) Hence prove (10.3.4) in the finite case.
10.4
Dual polar spaces
We now look at polar spaces “the other way up”. That is, given an abstract polar space of polar rank n, we consider the geometry whose POINTs and LINEs are
the subspaces of dimension n 1 and n 2 respectively, incidence being reversed
inclusion. (This geometry was introduced in Section 7.4.)
Proposition 10.8 A dual polar space of rank n is a near 2n-gon.
Proof This is implicit in what we proved in Proposition 7.9.
Any dual polar space has girth 4, and any circuit of length 4 is contained
in a unique quad. Moreover, the point-quad pairs are all classical. Both these
assertions are easily checked in the polar space by factoring out the intersection
of the subspaces in question.
The converse of this result was proved by Cameron . It is stated here using
the notation and ideas (and simplifications) of Shult and Yanushka described in
the last section.
Theorem 10.9 Let 9 be a near 2n-gon. Suppose that
156
10. Exterior powers and Clifford algebras
(a) any 4-circuit is contained in a quad;
(b) any point-quad pair is classical;
(c) chains of subspaces are finite.
Then 9 is a dual polar space of rank n.
Proof The ideas behind the proof will be sketched.
Given a point p, the residue of p (that is, the geometry of lines and quads
containing p) is a linear space, by hypothesis (a). Using (b), it is possible to show
that this linear space satisfies the Veblen–Young axioms, and so is a projective
space : p (possibly infinite-dimensional). We may assume that this geometry
has dimension greater than 2 (otherwise
the next few steps are vacuous).
Now,
given points p and q, let ; p q be the set of lines through p (i.e., points
of : p ) which belong to geodesics from p to q (that is, which contain points r
with d q r d p q < 1). This set is a subspace of : p . Let X be any subspace
=
of : p , and let
p X q : ; p q >- X =
=
It can be shown that
p X is a subspace of the geometry, containing
all geodesics
between any two of its points, and= that,
if
p
p
? is= any point of
X , then there is
a subspace X ? of : p? such that
p?@ X ? p X= . For the final step, it is shown that the subspaces
p X , ordered by reverse
inclusion, satisfy the axioms (P1)–(P4) of Tits.
Remark In the case when any line has more than two points, condition (a) is a
consequence of (10.3.2), and (10.3.4) shows that (b) is equivalent to the assertion
that no point-quad pairs are ovoidal.
10.5
Clifford algebras and spinors
Spinors provide projective embeddings of some geometries related to dual
polar spaces, much as exterior powers do for subspace geometries. But they are
somewhat elusive, and we have to construct them via Clifford algebras.
Let V be a vector space over a commutative field F, and f a quadratic form on
V ; let b be the bilinear
form obtained by polarising f . The Clifford algebra C f of f (or of the pair V f ) is the free-est algebra generated by V subject to the
10.5. Clifford algebras and spinors
157
condition that v2 f v A 1 for all v V . In other words, it is the quotient of the
tensor algebra T V by the ideal generated by all elements v2 f v 1 for v V .
Note that vw wv b v w B 1 for v w V .
The Clifford algebra is a generalisation of the exterior algebra, to which it
reduces if f is identically zero. And it has the same dimension:
Proposition 10.10 Let v1 vn be a basis for V . Then C f has a basis consisting of all vectors vi1 vik , for 0 i1 ik n; and so rk C f 2n .
Proof Any product of basis vectors can be rearranged into non-decreasing order,
modulo products of smaller numbers of basis vectors, using
wv vw b v w B 1 A product with two terms equal can have its length reduced. Now the result follows by multilinearity.
In an important special case, we can describe the structure of C f .
Theorem 10.11 Let f be a split quadratic form of rank n over F (equivalent to
x1 x2 x3 x4 CD x2n 3 1 x2n Then C f % M2n F , the algebra of 2n 2n matrices over F.
Proof It suffices to find a linear map θ : V E
M2n F satisfying
(a) θ V generates M2n F (as algebra with 1);
(b) θ v 2 f v I for all v V .
For if so, then M2n F is a homomorphic image of C f ; comparing dimensions,
they are equal.
We use induction on n. For n 0, the result is trivial.
Suppose that it is
true for n, with a map θ. Let Ṽ V FG/ x y 0 , where f λx µy λµ. Define
θ̃ : Ṽ E M2nH 1 F by
θ̃ v θ v
O
O
θ v v V
158
10. Exterior powers and Clifford algebras
O
θ̃ x O
O O
θ̃ y I O
I
O
extended linearly.
B
To show generation, let ! CA D
" M2n H 1 F be given. We may assume inductively that A B C D are linear combinations
of products of θ v , with v V . The
same combinations of products of θ̃ v have the forms Ã ! OA AOIJ" , etc. Now
B
Ãθ̃ x θ̃ y 7
B̃θ̃ x 7 θ̃ y C̃ θ̃ y D̃θ̃ x K
D
A
C
To establish the relations, we note that
θ v
θ̃ v λx µy µI
λI
θ v and the square of the right-hand side is f v 7 λµ ! IOOI " , as required.
More generally, the argument shows the following.
Theorem 10.12 If the quadratic form f has rank n and germ f0 , then
C f % C f0 F
M2n F K
In
particular, C x20 x1 x2 L x2n 3 1 x2n is the direct sum of two copies of
M2n F ; and, if α is a non- square in F, then
NM
C αx20 x1 x2 CD x2n 3 1 x2n % M2n K where K F α .
Looked at more abstractly, Theorem 10.12 says that the Clifford algebra of
the split form of rank n is isomorphic to the algebra of endomorphisms of a vector
space S of rank 2n . This space is the spinor space, and its elements are called
spinors. Note that the connection between the spinor space and the original vector
space is somewhat abstract and tenuous! It is the spinor space which carries the
geometrical structures we now investigate.
Exercise
1. Prove that the Clifford algebras of the real quadratic forms x2 and x2 y2
respectively are isomorphic to the complex numbers and the quaternions. What is
the Clifford algebra of x2 y2 z2 ?
10.6. The geometry of spinors
10.6
159
The geometry of spinors
In order to connect spinors to the geometry of the quadratic form, we first need
to recognise the points of a vector space within its algebra of endomorphisms.
let V be a vector space, A the algebra of linear transformations of V . Then A
is a simple algebra. If U is any subspace of V , then
I U a A : va U for all v V is a left ideal in A. Every left ideal is of this form (see Exercise 1). So the projective space based on V is isomorphic to the lattice of left ideals of A. In particular,
the minimal left ideals correspond
to the points of the projective space. Moreover,
if U has rank 1, then I U has rank n, and A (acting by left multiplication) induces the algebra of linear transformations of U . In this way, the vector space is
“internalised” in the algebra.
Now let V carry a split quadratic form of rank n. If U is a totally singular
subspace of rank n, then the elements of U generate a subalgebra isomorphic to
the exterior algebra of U . Let Û denote the product of the vectors in a basis of U .
Note that Û is unchanged, apart from a scalar factor, if a different basis is used.
Then vuÛ 0 whenever v V , u U , and u 0; so the left ideal generated by
Û has dimension 2n (with a basis of the form vi1 vik Û , where v1 vn is
a basis of a complement
for U , and 1 i1 ik n. Thus, Û generates a
minimal left ideal of C f . By the preceding paragraph, this ideal corresponds to
a point of the projective space PG 2n 1 F based on the spinor space S.
Summarising, we have a map from the maximal totally singular subspaces of
the hyperbolic quadric to a subset of the points of projective spinor space. The
elements in the image of this map are called pure spinors.
We now state some properties of pure spinors without proof.
Proposition 10.13 (a) There is a decomposition of the spinor space S into two
subspaces S , S 3 , each of rank 2n 3 1 . Any pure spinor is contained in one
of these subspaces.
(b) Any line of spinor space which contains more than two pure spinors has the
form
/ Û 0 : U is t.s. with rank n U has type ε U O W where W is a t.s. subspace of rank n 2, and ε QP 1.
160
10. Exterior powers and Clifford algebras
In (a), the subspaces S and S 3 are called half-spinor spaces.
In (b), the type of a maximal t.s. subspace is that described in Section 7.4. The
maximal t.s. subspaces containing W form a dual polar space of rank 2, which in
this case is simply a complete bipartite graph, the parts of the bipartition being
the two types of maximal subspace. Any two subspaces of the same type have
intersection with even codimension at most 2, and hence intersect precisely in W .
The dual polar space associated with the split quadratic form has two points
per line, and so in general is a bipartite graph. The two parts of the bipartition
can be identified with the pure spinors in the two half-spinor spaces. The lines
described in (b) within each half-spinor space form a geometry, a so-called halfspinor geometry: two pure spinors are collinear in this geometry if and only if they
lie at distance 2 in the dual polar space. In general, distances in the half-spinor
geometry are those in the dual polar space, halved!
Proposition 10.14 If p is a point and L a line in a half-spinor geometry, then
either there is a unique point of L nearest p, or all points of L are equidistant from
p.
Proof Recall that the line L of the half-spinor geometry is “half” of a complete
bipartite graph Q, which is a quad in the dual polar space. If the gateway to Q is
on L, it is the point of L nearest to p; if it is on the other side, then all points of L
are equidistant from p.
The cases n 3 4 give us yet another way of looking at the Klein quadric and
triality.
Example n 3. The half-spinor space has rank 4. The diameter
of the half
spinor geometry is 1, and so it is a linear space; necessarily PG 3 F : that is,
every spinor in the half-spinor space is pure. Points of this space correspond to
one family of maximal subspaces on the Klein quadric.
Example n 4. Now the half-spinor spaces have rank 8, the same as V . The
half-spinor space has diameter 2, and (by Proposition 10.14) satisfies the Buekenhout–
Shult axiom. But we do not need to use the full classification of polar spaces here,
since the geometry is already embedded in PG 7 F ! We conclude that each halfspinor space is isomorphic to the original hyperbolic quadric.
We conclude by embedding a couple more dual polar spaces in projective
spaces.
10.6. The geometry of spinors
161
Proposition 10.15 Let f be a quadratic form of rank n 1 on a vector space of
rank 2n 1. Then the dual polar space of F is embedded as all the points and
some of the lines of the half-spinor space associated with a split quadratic form
of rank n.
Proof We can regard the given space as of the form v R , where v is a non-singular
vector in a space carrying a split quadratic form of rank n. Now each t.s. subspace
of rank n 1 for the given form is contained in a unique t.s. space of rank n of each
type for the split form; so we have an injection from the given dual polar space to
a half-spinor space. The map is onto: for if U is t.s. of rank n, then U . cR has
rank n 1. A line of the dual polar space consists of all the subspaces containing
a fixed t.s. subspace of rank n 2, and so translates into a line of the half-spinor
space, as required.
Proposition 10.16 Let K be a quadratic extension of F, with Galois automorphism σ. Let V be a vector space of rank 2n over K, carrying a non-degenerate
σ-Hermitian form b of rank n. Then the dual polar space associated with b is
embeddable in a half-spinor geometry over F.
Proof Let H v b v v . Then H v F for all v V ; and H is a quadratic
form on the space VF obtained by restricting scalars to F. (Note that VF has rank
4n over F.) Now any maximal t.i. subspace for b is a maximal t.s. subspace for
H of rank 2n; so H is a split form, and we have an injection from points of the
dual unitary space to pure spinors. Moreover, the intersection of any two of these
maximal t.s. subspaces has even F-codimension in each; so they all have the same
type, and our map goes to points of a half-spinor geometry.
A line of the dual polar space is defined by a t.i. subspace of rank n 1 (over
K), which is t.s. of rank 2n 2 over F; so it maps to a line of the half-spinor
geometry, as required.
In the case n 3, we have the duality between the unitary and non-split orthogonal spaces discussed in Section 8.3.
Exercise
1. (a) Prove that the set of endomorphisms of V with range contained in a
subspace U is a left ideal.
(b) Prove that, if T has range U , then any endomorphism whose range is
contained in U is a left multiple of T .
162
10. Exterior powers and Clifford algebras
(c) Deduce that every left ideal of the endomorphism ring of V is of the
form described in (a).
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Index
abstract polar space, 105
addition, 1, 10, 32
affine plane, 21, 40
affine space, 3, 6
algebraic curve, 52
algebraic variety, 52
alternating bilinear form, 77
alternating groups, 131
alternative division rings, 114
anisotropic, 84
anti-automorphism, 2
atom, 39
atomic lattice, 39
automorphism, 17
axis, 61
chamber-connected, 70
characteristic, 1
Chevalley group, 133
circle, 68
classical groups, 114, 131
classical point-quad pair, 154
classical polar space, 88
Clifford algebra, 156
code, 34
coding theory, 34
collineation, 8
commutative field, 1, 147
complete bipartite graph, 97
complete graph, 68
configuration theorem, 23
conic, 52, 127
connected geometry, 66
coordinatisation, 9
corank, 66
coset geometry, 70
cospread, 47
cotype, 66
cross ratio, 59
cross-ratio, 8
Baer subplane, 141
bilinear form, 76
binary Golay code, 36
bispread, 47, 127
bits, 34
block, 16
Buekenhout geometry, 65
buildings, 131
bundle theorem, 58
degenerate conic, 55
derivation, 46
Desargues’ Theorem, 4, 22, 38
Desargues’ theorem, 133
Desarguesian planes, 24
Desarguesian spread, 46
Cayley numbers, 114
central collineation, 23
central collineations, 133
centre, 61
chamber, 70
168
INDEX
design, 16
determinant, 149
diagram, 67
digon, 67
dimension, 3
division ring, 1
dodecad, 140, 144
doubly-even self-dual code, 130
dual polar space, 107, 133
dual space, 3
duality, 75
duality principle, 6
egglike inversive plane, 58
elation, 23, 61
elliptic quadrics, 57
equianharmonic, 60
error-correcting codes, 34
exterior algebra, 149
exterior points, 53
exterior power, 148
exterior set, 103
exterior square, 148
Feit–Higman theorem, 132
field, 1
finite field, 1, 14
finite simple groups, 131
firm, 66
fixed field, 81
flag, 66
flag geometry, 132
flat, 3
flat C3 -geometry, 102
free plane, 20
Friendship Theorem, 22
Fundamental Theorem of Projective
Geometry, 8, 114
169
Galois’ Theorem, 2
gamma space, 150
Gaussian coefficient, 15
general linear group, 8
generalised polygons, 131
generalised projective plane, 68
generalised projective space, 39
generalised quadrangle, 90, 97
geometry, 65
germ, 85, 90, 92
ghost node, 142
Golay code, 137, 144
GQ, 98
graph
complete bipartite, 97
grid, 91, 98
group, 8
groups of Lie type, 131
Hadamard matrix, 144
half-spinor geometry, 160
half-spinor spaces, 160
Hamming codes, 35
Hamming distance, 34
harmonic, 60
Hermitian form, 77
Hessian configuration, 141
homology, 23, 63
hyperbolic line, 84
hyperbolic quadric, 91, 103
hyperoval, 57, 101, 141
hyperplane, 3, 33, 90
hyperplane at infinity, 3, 7
ideal hyperplane, 7
incidence relation, 65
interior points, 53
inversive plane, 58
170
isomorphism, 8
join, 39
Kirkman’s schoolgirl problem, 119
Klein quadric, 118
lattice, 39
left vector space, 2
Lie algebra, 133
line, 3, 89, 150
line at infinity, 21
linear code, 137
linear codes, 35
linear diagram, 69
linear groups, 131
linear space, 36, 40, 68
linear transformations, 8
Mathieu group, 139, 140, 143
matrix, 3, 18
meet, 39
Miquel’s theorem, 58
modular lattice, 39
Moufang condition, 114, 133
Moulton plane, 20
multiplication, 1, 10
multiply transitive group, 11
near polygon, 153
Neumaier’s geometry, 101
non-degenerate, 76
non-singular, 83
nucleus, 55
octad, 140
octonions, 114
opposite field, 2
opposite regulus, 46
order, 19, 22, 58
INDEX
orders, 98
orthogonal groups, 114
orthogonal space, 88
overall parity check, 137
ovoid, 57, 58, 125, 131, 135, 154
ovoidal point-quad pair, 154
Pappian planes, 25
Pappus’ Theorem, 24, 55
parallel, 7
parallel postulate, 21
parallelism, 41
parameters, 69
partial linear space, 67
Pascal’s Theorem, 55
Pascalian hexagon, 55
passant, 57
perfect, 83
perfect codes, 35
perfect field, 123
perspectivity, 27
plane, 3, 89
Playfair’s Axiom, 21, 41
point, 3, 16, 89
point at infinity, 21
point-shadow, 69
polar rank, 85, 88
polar space, 88
polarisation, 82
polarity, 77
prders, 69
prime field, 1
probability, 17
projective plane, 4, 19, 40, 68
projective space, 3
projectivity, 27
pseudoquadratic form, 82, 113
pure products, 151
INDEX
pure spinors, 159
quad, 153
quadratic form, 82
quaternions, 2
radical, 80, 110
rank, 3, 66, 89
reduced echelon form, 18
Ree group, 135
Reed–Muller code, 139
reflexive, 77
regular spread, 46, 127
regulus, 46, 127
residually connected geometry, 66
residue, 66
right vector space, 2
ruled quadric, 91
Schläfli configuration, 100
secant, 57
Segre’s Theorem, 52
semilinear transformations, 8
semilinear, 76
sesquilinear, 76
shadow, 69
sharply t-transitive, 28
singular subspace, 105
skew field, 1
solid, 41
solids, 128
spinor space, 158
spinors, 158
sporadic groups, 131
spread, 45, 125, 127, 131
Steiner quadruple system, 43
Steiner triple system, 43
subspace, 33, 36, 90, 105
subspace geometry, 150
171
sum of linear spaces, 38
support, 35, 140
Suzuki–Tits ovoids, 58
symmetric algebra, 149
symmetric bilinear form, 77
symmetric power, 149
symmetric square, 149
symplectic groups, 114
symplectic space, 88
symplectic spread, 128
t.i. subspace, 88
t.s. subspace, 88
tangent, 57
tangent plane, 57
tensor algebra, 148
tensor product, 147
tetrad, 142
theory of perspective, 7
thick, 37, 66
thin, 66
totally isotropic subspace, 88
totally singular subspace, 88
trace, 81
trace-valued, 113
trace-valued Hermitian form, 81
transitivity of parallelism, 41
translation plane, 45
transvection, 61
transversality condition, 66
triality, 129, 133
triality quadric, 133
triangle property, 110
trio, 140
type map, 65
types, 65
unital, 141
172
unitary groups, 114
unitary space, 88
varieties, 65
variety, 16
Veblen’s Axiom, 4, 37, 42
Veblen’s axiom, 32, 40
Wedderburn’s Theorem, 1, 25
weight, 35
Witt index, 85
Witt system, 140
INDEX
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